Geodesic
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
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 2028-2044

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A number of properties of pseudo-vertices of a boundary value set in the Dirichlet problem to the first-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 pseudo-vertices 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 pseudo-vertices 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.
Mots-clés : eikonal
Keywords: Hamilton–Jacobi equation, minimax solution, velocity, diffeomorohism, optimal result function, singular set, symmetry, transversality. 
@article{SEMR_2020_17_a109,
     author = {A. A. Uspenskii and P. D. Lebedev},
     title = {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},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {2028--2044},
     publisher = {mathdoc},
     volume = {17},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2020_17_a109/}
}
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A. A. Uspenskii; P. D. Lebedev. 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. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 2028-2044. http://geodesic.mathdoc.fr/item/SEMR_2020_17_a109/