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This article is cited in 1 scientific paper (total in 1 paper)
Differentical equations, dynamical systems and optimal control
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
A. A. Uspenskiiab, P. D. Lebedevab a Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 16, S. Kovalevskaya str., Yekaterinburg, 620990, Russia
b Ural Federal University, 19, Mira str., Yekaterinburg, 620002, Russia
Abstract:
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.
Keywords:
eikonal, Hamilton–Jacobi equation, minimax solution, velocity, diffeomorohism, optimal result function, singular set, symmetry, transversality.
Received May 30, 2020, published December 9, 2020
Citation:
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”, Sib. Èlektron. Mat. Izv., 17 (2020), 2028–2044
Linking options:
https://www.mathnet.ru/eng/semr1329 https://www.mathnet.ru/eng/semr/v17/p2028
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