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Trudy Inst. Mat. i Mekh. UrO RAN, 2009, Volume 15, Number 3, Pages 202–218 (Mi timm416)  

This article is cited in 6 scientific papers (total in 6 papers)

On the structure of locally Lipschitz minimax solutions of the Hamilton–Jacobi–Bellman equation in terms of classical characteristics

N. N. Subbotina, E. A. Kolpakova

Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences

Abstract: Necessary and sufficient conditions for the minimax solution to the Cauchy problem for the Hamilton–Jacobi–Bellman equation are obtained as viability conditions for classical characteristics inside the graph of the minimax solution. Using this property, a representative formula for a one-dimensional conservation law in terms of classical characteristics is derived. An estimate of the numerical integration of the characteristic system is presented and errors of numerical realizations of representative formulas are determined for the conservation law and its potential equal to the minimax solution of the Hamilton–Jacobi–Bellman equation.

Keywords: Hamilton–Jacobi–Bellman equations, minimax/viscosity solutions, conservation laws, entropy solutions, method of characteristics.

Full text: PDF file (314 kB)
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English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2010, 268, suppl. 1, S222–S239

Bibliographic databases:

UDC: 519.857
Received: 29.10.2008

Citation: N. N. Subbotina, E. A. Kolpakova, “On the structure of locally Lipschitz minimax solutions of the Hamilton–Jacobi–Bellman equation in terms of classical characteristics”, Trudy Inst. Mat. i Mekh. UrO RAN, 15, no. 3, 2009, 202–218; Proc. Steklov Inst. Math. (Suppl.), 268, suppl. 1 (2010), S222–S239

Citation in format AMSBIB
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\by N.~N.~Subbotina, E.~A.~Kolpakova
\paper On the structure of locally Lipschitz minimax solutions of the Hamilton--Jacobi--Bellman equation in terms of classical characteristics
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2009
\vol 15
\issue 3
\pages 202--218
\mathnet{http://mi.mathnet.ru/timm416}
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\transl
\jour Proc. Steklov Inst. Math. (Suppl.)
\yr 2010
\vol 268
\issue , suppl. 1
\pages S222--S239
\crossref{https://doi.org/10.1134/S0081543810050160}
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. E. A. Kolpakova, “Obobschennyi metod kharakteristik v teorii uravnenii Gamiltona–Yakobi i zakonov sokhraneniya”, Tr. IMM UrO RAN, 16, no. 5, 2010, 95–102  mathnet  elib
    2. N. N. Subbotina, E. A. Kolpakova, “Method of characteristics for optimal control problems and conservation laws”, Journal of Mathematical Sciences, 199:5 (2014), 588–595  mathnet  crossref  mathscinet
    3. 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  mathnet  mathscinet  elib
    4. A. S. Rodin, “O strukture singulyarnogo mnozhestva kusochno-gladkogo minimaksnogo resheniya uravneniya Gamiltona — Yakobi — Bellmana”, Tr. IMM UrO RAN, 21, no. 2, 2015, 198–205  mathnet  mathscinet  elib
    5. 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  mathnet  crossref  mathscinet  isi  elib
    6. Rodin A.S. Shagalova L.G., “Bifurcation Points of the Generalized Solution of the Hamilton-Jacobi-Bellman Equation”, IFAC PAPERSONLINE, 51:32 (2018), 866–870  crossref  isi  scopus
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