N. N. Subbotina, L. G. Shagalova, “On a solution to the Cauchy problem for the Hamilton–Jacobi equation with state constraints”, Trudy Inst. Mat. i Mekh. UrO RAN, 17, no. 2, 2011, 191

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Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2011, Volume 17, Number 2, Pages 191–208 (Mi timm707)

This article is cited in 12 scientific papers (total in 12 papers)

On a solution to the Cauchy problem for the Hamilton–Jacobi equation with state constraints

N. N. Subbotina, L. G. Shagalova

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

Full-text PDF (315 kB) Citations (12)

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Abstract: The Cauchy problem for the Hamilton–Jacobi equation, which appears in molecular biology for the Crow–Kimura model of molecular evolution, is considered. The notion of a continuous generalized solution of this problem with state constraints is introduced on the basis of the viscosity (and/or minimax) approach. A construction of the generalized solution of the problem is proposed, which uses the value function in an auxiliary optimal control problem with a given target set. It is shown that the generalized solution in the considered problem is not unique. The research is based on the generalized method of characteristics for the Hamilton–Jacobi equations in the Dirichlet problem.

Keywords: Hamilton–Jacobi equations, method of characteristics, viscosity solutions, minimax solutions, optimal control, value function.

Received: 06.12.2010

Bibliographic databases:

Document Type: Article

UDC: 517.95+517.977

Language: Russian

Citation: N. N. Subbotina, L. G. Shagalova, “On a solution to the Cauchy problem for the Hamilton–Jacobi equation with state constraints”, Trudy Inst. Mat. i Mekh. UrO RAN, 17, no. 2, 2011, 191–208

Citation in format AMSBIB

\Bibitem{SubSha11}

\by N.~N.~Subbotina, L.~G.~Shagalova

\paper On a~solution to the Cauchy problem for the Hamilton--Jacobi equation with state constraints

\serial Trudy Inst. Mat. i Mekh. UrO RAN

\yr 2011

\vol 17

\issue 2

\pages 191--208

\mathnet{http://mi.mathnet.ru/timm707}

\elib{https://elibrary.ru/item.asp?id=16352403}

Linking options:

https://www.mathnet.ru/eng/timm707

https://www.mathnet.ru/eng/timm/v17/i2/p191

This publication is cited in the following 12 articles:

Vladimir N. Ushakov, Aleksandr M. Tarasev, “Igrovaya zadacha sblizheniya nelineinoi upravlyaemoi sistemy”, MTIP, 15:2 (2023), 122–139
L. G. Shagalova, “Nepreryvnoe obobschennoe reshenie uravneniya Gamiltona – Yakobi s trekhkomponentnym gamiltonianom”, Tr. IMM UrO RAN, 28, no. 1, 2022, 257–268
L. G. Shagalova, “Obobschennoe reshenie uravneniya Gamiltona—Yakobi s trekhkomponentnym gamiltonianom”, Algebra, geometriya, differentsialnye uravneniya, Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 217, VINITI RAN, M., 2022, 63–72
L. G. Shagalova, “Nepreryvnoe obobschennoe reshenie uravneniya Gamiltona—Yakobi s nekoertsitivnym gamiltonianom”, Materialy Vserossiiskoi nauchnoi konferentsii «Differentsialnye uravneniya i ikh prilozheniya», posvyaschennoi 85-letiyu professora M. T. Terekhina. Ryazanskii gosudarstvennyi universitet im. S.A. Esenina, Ryazan, 17–18 maya 2019 g. Chast 2, Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 186, VINITI RAN, M., 2020, 144–151
Subbotina N.N. Shagalova L.G., “The Study of Evolution in the Crow - Kimura Molecular Genetics Model Using Methods of Calculus of Variations”, Proceedings of the 8th International Conference on Mathematical Modeling (ICMM-2017), AIP Conference Proceedings, 1907, ed. Egorov I. Popov S. Vabishchevich P. Antonov M. Lazarev N. Troeva M. Troeva M. Ivanova A. Grigorev Y., Amer Inst Physics, 2017, UNSP 020001
N. N. Subbotina, L. G. Shagalova, “On the continuous extension of a generalized solution of the Hamilton-Jacobi equation by characteristics that form a central field of extremals”, Proc. Steklov Inst. Math. (Suppl.), 293, suppl. 1 (2016), 183–198
N. N. Subbotina, L. G. Shagalova, “Konstruktsiya nepreryvnogo obobschennogo resheniya uravneniya Gamiltona–Yakobi s fazovymi ogranicheniyami”, Izv. IMI UdGU, 2015, no. 2(46), 193–201
N. N. Subbotina, L. G. Shagalova, “Konstruktsiya nepreryvnogo minimaksnogo/vyazkostnogo resheniya uravneniya Gamiltona–Yakobi–Bellmana s neprodolzhimymi kharakteristikami”, Tr. IMM UrO RAN, 20, no. 4, 2014, 247–257
L. G. Shagalova, “On the Solution of the Hamilton–Jacobi Equation Emerging in Molecular Biology”, Autom. Remote Control, 74:8 (2013), 1366–1377
N. N. Subbotina, L. G. Shagalova, “Postroenie obobschennogo resheniya uravneniya Gamiltona–Yakobi v ogranichennoi oblasti”, Izv. IMI UdGU, 2012, no. 1(39), 126–127
N. N. Subbotina, L. G. Shagalova, “Construction of a generalized solution to an equation that preserves the Bellman type in a given domain of the state space”, Proc. Steklov Inst. Math., 277 (2012), 234–247
Subbotina N.N., Shagalova L.G., “Primenenie teorii optimalnogo upravleniya dlya resheniya uravnenii gamiltona-yakobi s fazovymi ogranicheniyami”, Vestnik Tambovskogo universiteta. Seriya: Estestvennye i tekhnicheskie nauki, 16:4 (2011), 1185–1187

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Trudy Instituta Matematiki i Mekhaniki UrO RAN

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