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Mat. Sb., 1998, Volume 189, Number 11, Pages 27–60 (Mi msb364)  

This article is cited in 21 scientific papers (total in 21 papers)

Asymptotic behaviour of bounded controls for a singular elliptic problem in a domain with a small cavity

A. R. Danilin

Urals State Pedagogical University

Abstract: The problem of optimal control of solutions of an elliptic equation in a domain with a small cavity is discussed. Uniform asymptotic formulae for the solutions are obtained up to an arbitrary degree of the small parameter by the method of matching asymptotic expansions. Other aspects of the same problem have been considered by Kapustyan.


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English version:
Sbornik: Mathematics, 1998, 189:11, 1611–1642

Bibliographic databases:

UDC: 517.977
MSC: 35C20, 49K22
Received: 20.08.1997

Citation: A. R. Danilin, “Asymptotic behaviour of bounded controls for a singular elliptic problem in a domain with a small cavity”, Mat. Sb., 189:11 (1998), 27–60; Sb. Math., 189:11 (1998), 1611–1642

Citation in format AMSBIB
\by A.~R.~Danilin
\paper Asymptotic behaviour of bounded controls for a~singular elliptic problem in a~domain with a~small cavity
\jour Mat. Sb.
\yr 1998
\vol 189
\issue 11
\pages 27--60
\jour Sb. Math.
\yr 1998
\vol 189
\issue 11
\pages 1611--1642

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    This publication is cited in the following articles:
    1. Danilin, AR, “Asymptotics of control for a singular elliptic problem”, Doklady Akademii Nauk, 369:3 (1999), 305  mathnet  mathscinet  zmath  isi
    2. A. R. Danilin, “Approximation of a singularly perturbed elliptic problem of optimal control”, Sb. Math., 191:10 (2000), 1421–1431  mathnet  crossref  crossref  mathscinet  zmath  isi
    3. A. R. Danilin, “Asymptotic behaviour of solutions of a singular elliptic system in a rectangle”, Sb. Math., 194:1 (2003), 31–61  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    4. S. V. Zakharov, “Heat Distribution in an Infinite Rod”, Math. Notes, 80:3 (2006), 366–371  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    5. Danilin, AR, “Asymptotic behavior of the solution to the Cauchy problem for a Hamilton-Jacoby equation depending on a small parameter”, Doklady Mathematics, 73:2 (2006), 214  crossref  zmath  isi  elib  scopus  scopus  scopus
    6. Danilin, AR, “Asymptotics of the optimal value of the performance functional for a rapidly stabilizing indirect control in the regular case”, Differential Equations, 42:11 (2006), 1545  mathnet  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    7. M. G. Dmitriev, G. A. Kurina, “Singular perturbations in control problems”, Autom. Remote Control, 67:1 (2006), 1–43  mathnet  crossref  mathscinet  zmath  elib  elib
    8. A. R. Danilin, Yu. V. Parysheva, “The asymptotics of the optimal value of the performance functional in a linear optimal control problem in the regular case”, Proc. Steklov Inst. Math. (Suppl.), 259, suppl. 2 (2007), S83–S94  mathnet  crossref  elib
    9. A. R. Danilin, A. P. Zorin, “Asymptotics of a solution to an optimal boundary control problem”, Proc. Steklov Inst. Math. (Suppl.), 269, suppl. 1 (2010), S81–S94  mathnet  crossref  elib
    10. Danilin A.R., Zorin A.P., “Asymptotic Expansion of Solutions to Optimal Boundary Control Problems”, Doklady Mathematics, 84:2 (2011), 665–668  crossref  mathscinet  zmath  isi  elib  elib  scopus
    11. A. R. Danilin, “Optimalnoe granichnoe upravlenie v oblasti s maloi polostyu”, Ufimsk. matem. zhurn., 4:2 (2012), 87–100  mathnet
    12. A. R. Danilin, O. O. Kovrizhnykh, “Asymptotic representation of a solution to a singular perturbation linear time-optimal problem”, Proc. Steklov Inst. Math. (Suppl.), 281, suppl. 1 (2013), 22–35  mathnet  crossref  isi  elib
    13. A. R. Danilin, A. P. Zorin, “Asimptotika resheniya zadachi optimalnogo granichnogo upravleniya v ogranichennoi oblasti”, Tr. IMM UrO RAN, 18, no. 3, 2012, 75–82  mathnet  elib
    14. A. P. Zorin, “Asimptoticheskoe razlozhenie resheniya zadachi optimalnogo upravleniya ogranichennym potokom na granitse”, Tr. IMM UrO RAN, 19, no. 1, 2013, 115–120  mathnet  mathscinet  elib
    15. A. R. Danilin, “Solution asymptotics in a problem of optimal boundary control of a flow through a part of the boundary”, Proc. Steklov Inst. Math. (Suppl.), 292, suppl. 1 (2016), 55–66  mathnet  crossref  mathscinet  isi  elib
    16. A. R. Danilin, O. O. Kovrizhnykh, “Asimptotika optimalnogo vremeni v odnoi zadache o bystrodeistvii s malym parametrom”, Tr. IMM UrO RAN, 21, no. 1, 2015, 71–80  mathnet  mathscinet  elib
    17. S. V. Zakharov, “Asymptotic calculation of the heat distribution on a plane”, Proc. Steklov Inst. Math. (Suppl.), 296, suppl. 1 (2017), 243–249  mathnet  crossref  mathscinet  isi  elib
    18. A. R. Danilin, “Asimptoticheskoe razlozhenie resheniya singulyarno vozmuschennoi zadachi optimalnogo upravleniya s malym koeffitsientom koertsitivnosti”, Tr. IMM UrO RAN, 24, no. 3, 2018, 51–61  mathnet  crossref  elib
    19. A. R. Danilin, “Asymptotics of the solution of a bisingular optimal boundary control problem in a bounded domain”, Comput. Math. Math. Phys., 58:11 (2018), 1737–1747  mathnet  crossref  crossref  isi  elib
    20. Zakharov S.V., “Long-Time Behavior of the Solution of the Cauchy Problem For the Third-Order Airy Equation”, Asymptotic Anal., 116:2 (2020), 139–148  crossref  isi
    21. Danilin A.R., “Asymptotics of the Solution of a Singular Optimal Distributed Control Problem With Essential Constraints in a Convex Domain”, Differ. Equ., 56:2 (2020), 251–263  crossref  isi
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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