Construction of solution for optimal-time problem under variable border smoothness for nonconvex target set*

The paper deals with the problem of emergence of singular sets in the Dirichlet boundary value problem for the Hamiltonian type equations. Analytical and numerical procedures are proposed for constructing a generalized (minimax) solution of the Hamilton-Jacobi-Bellman equation. This solution is the function of optimal result for the corresponding optimal-time control problem. In particular, the subject of analysis is pseudo-vertices of a boundary target set. Search of pseudo-vertices is an element of the procedure for constructing branches of a singular set for the function of optimal result. Necessary conditions for existence of pseudo-vertices are given in the smooth case and in the case of weakened assumptions on differentiability of the boundary of a non-convex target set. Necessary conditions are formulated by means of stationarity of coordinate functions and in terms of one-sided curvatures. Examples are provided to illustrate the efficiency of the method.