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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.
DOI:
https://doi.org/10.4213/sm364
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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
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http://mi.mathnet.ru/eng/msb364https://doi.org/10.4213/sm364 http://mi.mathnet.ru/eng/msb/v189/i11/p27
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This publication is cited in the following articles:
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Danilin, AR, “Asymptotics of control for a singular elliptic problem”, Doklady Akademii Nauk, 369:3 (1999), 305
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A. R. Danilin, “Approximation of a singularly perturbed elliptic problem of optimal control”, Sb. Math., 191:10 (2000), 1421–1431
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A. R. Danilin, “Asymptotic behaviour of solutions of a singular elliptic system
in a rectangle”, Sb. Math., 194:1 (2003), 31–61
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S. V. Zakharov, “Heat Distribution in an Infinite Rod”, Math. Notes, 80:3 (2006), 366–371
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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
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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
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M. G. Dmitriev, G. A. Kurina, “Singular perturbations in control problems”, Autom. Remote Control, 67:1 (2006), 1–43
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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
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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
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Danilin A.R., Zorin A.P., “Asymptotic Expansion of Solutions to Optimal Boundary Control Problems”, Doklady Mathematics, 84:2 (2011), 665–668
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A. R. Danilin, “Optimalnoe granichnoe upravlenie v oblasti s maloi polostyu”, Ufimsk. matem. zhurn., 4:2 (2012), 87–100
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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
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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
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A. P. Zorin, “Asimptoticheskoe razlozhenie resheniya zadachi optimalnogo upravleniya ogranichennym potokom na granitse”, Tr. IMM UrO RAN, 19, no. 1, 2013, 115–120
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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
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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
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S. V. Zakharov, “Asymptotic calculation of the heat distribution on a plane”, Proc. Steklov Inst. Math. (Suppl.), 296, suppl. 1 (2017), 243–249
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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
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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
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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
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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
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