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Zh. Vychisl. Mat. Mat. Fiz., 2013, Volume 53, Number 1, Pages 20–46 (Mi zvmmf9791)  

This article is cited in 14 scientific papers (total in 14 papers)

Difference approximations of optimization problems for semilinear elliptic equations in a convex domain with controls in the coefficients multiplying the highest derivatives

F. V. Lubyshev, A. R. Manapova

Bashkir State University, Ufa

Abstract: Finite difference approximations are proposed for nonlinear optimal control problems for a non-self-adjoint elliptic equation with Dirichlet boundary conditions in a convex domain $\Omega\subset\mathbb{R}^2$ with controls involved in the leading coefficients. The convergence of the approximations with respect to the state, functional, and control is analyzed, and a regularization of the approximations is proposed.

Key words: non-self-adjoint elliptic semilinear equations, control in the coefficients multiplying high-est derivatives, difference approximations, convergence of approximations.

DOI: https://doi.org/10.7868/S0044466913010079

Full text: PDF file (446 kB)
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English version:
Computational Mathematics and Mathematical Physics, 2013, 53:1, 8–33

Bibliographic databases:

UDC: 519.626
Received: 19.07.2012

Citation: F. V. Lubyshev, A. R. Manapova, “Difference approximations of optimization problems for semilinear elliptic equations in a convex domain with controls in the coefficients multiplying the highest derivatives”, Zh. Vychisl. Mat. Mat. Fiz., 53:1 (2013), 20–46; Comput. Math. Math. Phys., 53:1 (2013), 8–33

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. 51, no. 4, 2015, 548–557  crossref  mathscinet  zmath  isi  elib  scopus
    2. Andrei V. Chernov, “O suschestvovanii $\varepsilon$-ravnovesiya v differentsialnykh igrakh, svyazannykh s ellipticheskimi uravneniyami, upravlyaemymi mnogimi igrokami”, MTIP, 6:1 (2014), 91–115  mathnet
    3. A. R. Manapova, F. V. Lubyshev, “Accuracy estimate with respect to state of finite-dimensional approximations for optimization problems for semi-linear elliptic equations with discontinuous coefficients and solutions”, Ufa Math. J., 6:3 (2014), 69–84  mathnet  crossref  elib
    4. F. V. Lubyshev, A. R. Manapova, M. E. Fairuzov, “Approximations of optimal control problems for semilinear elliptic equations with discontinuous coefficients and solutions and with control in matching boundary conditions”, Comput. Math. Math. Phys., 54:11 (2014), 1700–1724  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    5. A. V. Chernov, “On the convergence of the conditional gradient method as applied to the optimization of an elliptic equation”, Comput. Math. Math. Phys., 55:2 (2015), 212–226  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    6. A. V. Chernov, “O kusochno postoyannoi approksimatsii v raspredelennykh zadachakh optimizatsii”, Tr. IMM UrO RAN, 21, no. 1, 2015, 264–279  mathnet  mathscinet  elib
    7. A. V. Chernov, “Ob analoge teoremy Uintnera dlya upravlyaemogo ellipticheskogo uravneniya”, Izv. IMI UdGU, 2015, no. 2(46), 228–235  mathnet  elib
    8. A. V. Chernov, “On the structure of a solution set of controlled initial-boundary value problems”, Russian Math. (Iz. VUZ), 60:2 (2016), 62–71  mathnet  crossref  isi
    9. A. R. Manapova, F. V. Lubyshev, “Numerical solution of optimization problems for semi-linear elliptic equations with discontinuous coefficients and solutions”, Appl. Numer. Math., 104:SI (2016), 182–203  crossref  mathscinet  zmath  isi  scopus
    10. F. V. Lubyshev, M. E. Fairuzov, “Approximations of optimal control problems for semilinear elliptic equations with discontinuous coefficients and states and with controls in the coefficients multiplying the highest derivatives”, Comput. Math. Math. Phys., 56:7 (2016), 1238–1263  mathnet  crossref  crossref  isi  elib
    11. A. V. Chernov, “Differentiation of a functional in the problem of parametric coefficient optimization in the global electric circuit equation”, Comput. Math. Math. Phys., 56:9 (2016), 1565–1579  mathnet  crossref  crossref  isi  elib
    12. A. V. Chernov, “Differentiation of the functional in a parametric optimization problem for a coefficient of a semilinear elliptic equation”, Differ. Equ., 53:4 (2017), 551–562  crossref  mathscinet  zmath  isi  elib  elib  scopus
    13. F. V. Lubyshev, A. R. Manapova, “Approksimatsiya zadach optimalnogo upravleniya koeffitsientami ellipticheskikh uravnenii konvektsii-diffuzii s usloviyami sopryazheniya tipa neidealnogo kontakta”, Zhurnal SVMO, 21:2 (2019), 187–214  mathnet  crossref  elib
    14. A. V. Chernov, “On differentiation of functional in problem on parametric coefficient optimization in semilinear global electric circuit equation”, Ufa Math. J., 13:3 (2021), 152–173  mathnet  crossref
  • Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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