Abstract:
We study the problem of the appearance of nonsmooth singularities in the evolution of plane wavefronts in the Dirichlet problem for a first-order partial differential equation. The approach to investigating the singularities is based on the properties of local diffeomorphisms. A generalization of the classical notion of a derivative is introduced, which coincides in particular cases with the Schwarz derivative. The results of modeling solutions of nonsmooth dynamic problems are presented.
Citation:
A. A. Uspenskii, P. D. Lebedev, “On the set of limit values of local diffeomorphisms in wavefront evolution”, Trudy Inst. Mat. i Mekh. UrO RAN, 16, no. 1, 2010, 171–185; Proc. Steklov Inst. Math. (Suppl.), 272, suppl. 1 (2011), S255–S270
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\by A.~A.~Uspenskii, P.~D.~Lebedev
\paper On the set of limit values of local diffeomorphisms in wavefront evolution
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2010
\vol 16
\issue 1
\pages 171--185
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\jour Proc. Steklov Inst. Math. (Suppl.)
\yr 2011
\vol 272
\issue , suppl. 1
\pages S255--S270
\crossref{https://doi.org/10.1134/S0081543811020180}
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Linking options:
https://www.mathnet.ru/eng/timm536
https://www.mathnet.ru/eng/timm/v16/i1/p171
This publication is cited in the following 16 articles:
A. A. Uspenskii, P. D. Lebedev, “Alpha sets and their hulls: analytical relationships in the plane case”, Russian Universities Reports. Mathematics, 29:146 (2024), 204–217
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 resheniya zadachi upravleniya po bystrodeistviyu pri narushenii gladkosti krivizny granitsy tselevogo mnozhestva”, Izv. IMI UdGU, 53 (2019), 98–114
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
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
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
A. A. Uspenskii, “Derivatives by virtue of diffeomorphisms and their applications in control theory and geometrical optics”, Proc. Steklov Inst. Math. (Suppl.), 293, suppl. 1 (2016), 238–253
A. A. Uspenskii, “Calculation formulas for nonsmooth singularities of the optimal result function in a time-optimal problem”, Proc. Steklov Inst. Math. (Suppl.), 291, suppl. 1 (2015), 239–254
V. N. Ushakov, A. A. Uspenskii, P. D. Lebedev, “Geometriya singulyarnykh krivykh dlya odnogo klassa zadach bystrodeistviya”, Vestn. S.-Peterburg. un-ta. Ser. 10. Prikl. matem. Inform. Prots. upr., 2013, no. 3, 157–167
A. A. Uspenskii, P. D. Lebedev, P. A. Vasev, “Approksimatsiya negladkoi funktsii optimalnogo rezultata v odnom klasse zadach bystrodeistviya”, Vestnik ChelGU, 2013, no. 16, 71–77
A. A. Uspenskii, P. D. Lebedev, “Algoritmy postroeniya singulyarnykh mnozhestv dlya odnogo klassa zadach bystrodeistviya”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2010, no. 3, 30–41
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