Calculation formulas for nonsmooth singularities of the optimal result function in a time-optimal problem

On singularity structure of minimax solution to Dirichlet problem for eikonal type equation with discontinuous curvature of boundary of boundary set

“…Here we mean the points (𝑡 2 , 𝑡 1 ) = (𝑡 0 , 𝑡 0 ) lying on the bisectrix of the first and third coordinate angles in the space of the parameters, which are limiting for the graphs of the local diffeomorphisms. The existence of such points and corresponding local diffeomorphisms was shown at examples in [12].…”

“…We mention that it is easiest to find the pseudo-vertices for two classes of curves enveloping the boundary set. The first class is formed by piece-wise smooth curves, the corner points of which are pseudo-vertices [12]. The second class is the curves with the smoothness of order at least three, the pseudo-vertices of which are contained in the set of the points with stationary curvature [13].…”

Combined Algorithms for Constructing a Solution to the Time–optimal Problem in Three-Dimensional Space Based on the Selection of Extreme Points of the Scattering Surface

“…It should be noted that (4.8) for t = t 0 can be represented as a set of points, for which the following conditions hold: 4.33) for the extreme points of a singular set are generalizations to three-dimensional equations for the extreme points of a singular set for solving the corresponding planar time-optimal control problem (see (4.1) and (4.2) from [18]).…”

“…In the problem under consideration, the bisector generally consists of the union of two-dimensional, one-dimensional, and zero-dimensional manifolds [11,12]. The simulation of the non-smooth solution of the problem was carried out with the help of modernized computational procedures, previously created for solving flat problems of time-optimal control [18]. The developed procedures can be used in constructing generalized solutions of first-order partial differential equations [15], as well as in theoretical mechanics, geometric optics, seismology, and economics [5].…”