Abstract:
A formula for a minimax (generalized) solution of the Cauchy–Dirichlet problem for an eikonal-type equation is proved in the case of an isotropic medium providing that the edge set is closed; the boundary of the edge set can be nonsmooth. A technique of constructing a minimax solution is proposed that uses methods from the theory of singularities of differentiable mappings. The notion of a bisector, which is a representative of symmetry sets, is introduced. Special points of the set boundary–pseudovertices–are singled out and bisector branches corresponding to them are constructed; the solution suffers a “gradient catastrophe” on these branches. Having constructed the bisector, one can generate the evolution of wave fronts in smoothness domains of the generalized solution. The relation of the problem under consideration to one class of time-optimal dynamic control problems is shown. The efficiency of the developed approach is illustrated by examples of analytical and numerical construction of minimax solutions.
Citation:
P. D. Lebedev, A. A. Uspenskii, V. N. Ushakov, “Construction of a minimax solution for an eikonal-type equation”, Trudy Inst. Mat. i Mekh. UrO RAN, 14, no. 2, 2008, 182–191; Proc. Steklov Inst. Math. (Suppl.), 263, suppl. 2 (2008), S191–S201
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\paper Construction of a~minimax solution for an eikonal-type equation
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\pages 182--191
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\jour Proc. Steklov Inst. Math. (Suppl.)
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Linking options:
https://www.mathnet.ru/eng/timm34
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This publication is cited in the following 33 articles:
P. D. Lebedev, A. A. Uspenskii, “Numerical-analytic construction of a generalized solution to the eikonal equation in the plane case”, Sb. Math., 215:9 (2024), 1224–1248
Pavel D. Lebedev, Alexander A. Uspenskii, “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”, Ural Math. J., 8:2 (2022), 115–126
A. A. Uspenskii, P. D. Lebedev, “O strukture singulyarnogo mnozhestva resheniya v odnom klasse prostranstvennykh zadach upravleniya po bystrodeistviyu”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 31:3 (2021), 471–486
Uspenskii A.A., Lebedev P.D., “On Singularity Structure of Minimax Solution to Dirichlet Problem For Eikonal Type Equation With Discontinuous Curvature of Boundary of Boundary Set”, Ufa Math. J., 13:3 (2021), 126–151
P. D. Lebedev, A. A. Uspenskii, “Postroenie rasseivayuschikh krivykh v odnom klasse zadach bystrodeistviya pri skachkakh krivizny granitsy tselevogo mnozhestva”, Izv. IMI UdGU, 55 (2020), 93–112
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Alimov A.R., “Singularities of Solutions of the Eikonal Equation”, Differ. Equ., 55:10 (2019), 1311–1316
A. A. Uspenskii, P. D. Lebedev, “Vyyavlenie singulyarnosti u obobschennogo resheniya zadachi Dirikhle dlya uravneniya tipa eikonala v usloviyakh minimalnoi gladkosti granitsy kraevogo mnozhestva”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 28:1 (2018), 59–73
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i ikh prilozheniya v teorii upravleniya”, Vestn. S.-Peterburg. un-ta. Ser. 10. Prikl. matem. Inform. Prots. upr., 14:3 (2018), 261–272
Lebedev P.D., Uspenskii A.A., “Construction of Singular Sets in a Velocity Control Problem With Nonconvex Target”, IFAC PAPERSONLINE, 51:32 (2018), 681–686
A. A. Uspenskii, P. D. Lebedev, “Evklidovo rasstoyanie do zamknutogo mnozhestva kak minimaksnoe reshenie zadachi Dirikhle dlya uravneniya Gamiltona-Yakobi”, Vestnik Tambovskogo universiteta. Seriya: estestvennye i tekhnicheskie nauki, 23:124 (2018), 797–804
Lebebev P.D., Uspenskii A.A., Ushakov V.N., “Construction of Nonsmooth Solutions in One Class of Velocity Problems”, 2017 Constructive Nonsmooth Analysis and Related Topics (Dedicated to the Memory of V. F. Demyanov) (CNSA), ed. Polyakova L., IEEE, 2017, 185–188
Pavel D. Lebebev, Aleksandr A. Uspenskii, Vladimir N. Ushakov, 2017 Constructive Nonsmooth Analysis and Related Topics (dedicated to the memory of V.F. Demyanov) (CNSA), 2017, 1
A. R. Alimov, I. G. Tsar'kov, “Connectedness and solarity in problems of best and near-best approximation”, Russian Math. Surveys, 71:1 (2016), 1–77
A. A. Uspenskii, P. D. Lebedev, “The construction of singular curves for generalized solutions of eikonal-type equations with a curvature break in the boundary of the boundary set”, Proc. Steklov Inst. Math. (Suppl.), 297, suppl. 1 (2017), 191–202
P. D. Lebedev, A. A. Uspenskii, “Postroenie funktsii optimalnogo rezultata i rasseivayuschikh linii v zadachakh bystrodeistviya s nevypuklym tselevym mnozhestvom”, Tr. IMM UrO RAN, 22, no. 2, 2016, 188–198
V. N. Ushakov, A. A. Uspenskii, “Theorems on the separability of α-sets in Euclidean space”, Proc. Steklov Inst. Math. (Suppl.), 299, suppl. 1 (2017), 231–245
Lebedev P.D., Tarasyev A.M., Uspenskii A.A., “Construction of Solution For Optimal-Time Problem Under Variable Border Smoothness For Nonconvex Target Set”, IFAC PAPERSONLINE, 49:18 (2016), 386–391
A. A. Uspenskii, “Neobkhodimye usloviya suschestvovaniya psevdovershin kraevogo mnozhestva v zadache Dirikhle dlya uravneniya eikonala”, Tr. IMM UrO RAN, 21, no. 1, 2015, 250–263
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