Far Eastern Mathematical Journal

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Optimal control of the radiation heat exchange equations for multi-component media


Chebotarev A.Yu.

2021, issue 1, P. 113-121
DOI: https://doi.org/10.47910/FEMJ202110


Abstract
An analysis of optimal control problems for nonlinear elliptic equations modeling complex heat transfer with Fresnel conjugation conditions on the discontinuity surfaces of the refractive index is presented. Conditions for the solvability of extremal problems and the nondegeneracy of the optimality system are obtained. For the control problem with boundary observation, the bang-bang property is set.

Keywords:
stationary equations of radiative heat transfer, Fresnel conjugation conditions,

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References

[1] Alexander Yu. Chebotarev and Gleb V. Grenkin and Andrey E. Kovtanyuk and Nikolai D. Botkin and Karl-Heinz Hoffmann, “Diffusion approximation of the radiative-conductive heat transfer model with Fresnel matching conditions”, Communications in Nonlinear Science and Numerical Simulation, 57 (2018), 290–298.
[2] R. Pinnau, “Analysis of Optimal Boundary Control for Radiative Heat Transfer Modelled by the SP1-System”, Comm. Math. Sci., 5:4 (2007), 951–969.
[3] A.E. Kovtanyuk, A.Yu. Chebotarev, N.D. Botkin, K.-H. Hoffmann, “The unique solvability of a complex 3D heat transfer problem”, J. Math. Anal. Appl., 409:2 (2014), 808–815.
[4] A.E. Kovtaniuk, A.Iu. Chebotarev, “Statsionarnaia zadacha slozhnogo teploobmena”, Zh. vychisl. matem. i matem. fiz., 54:4 (2014), 711–719.
[5] A.E. Kovtaniuk, A.Iu. Chebotarev, “Statsionarnaia zadacha svobodnoi konvektsii s radiatsionnym teploobmenom” , Differentsial'nye uravneniia, 50:12 (2014), 1590–1597.
[6] Kovtanyuk Andrey E., Chebotarev Alexander Yu., Botkin Nikolai D., and Hoffmann Karl-Heinz, “Theoretical analysis of an optimal control problem of conductive convective radiative heat transfer”, J. Math. Anal. Appl., 412 (2014), 520–528.
[7] A.E. Kovtanyuk, A.Yu. Chebotarev, N.D. Botkin, and Hoffman Karl-Heinz, “Unique solvability of a steady-state complex heat transfer model”, Commun. Nonlinear Sci. Numer. Simulat., 20 (2015), 776–784.
[8] G.V. Grenkin, A.Iu. Chebotarev, “Neodnorodnaia nestatsionarnaia zadacha slozhnogo teploobmena”, Sibirskie elektronnye matematicheskie izvestiia, 12:11 (2015), 562–576.
[9] G.V. Grenkin, A.Iu. Chebotarev, “Nestatsionarnaia zadacha svobodnoi konvektsii s radiatsionnym teploobmenom” , Zh. vychisl. matem. fiz., 56:2 (2016), 275–282.
[10] A. Chebotarev, A. Kovtanyuk, G. Grenkin, N. Botkin, and K.-H. Hoffman, “Boundary optimal control problem of complex heat transfer model”, J. Math. Anal. Appl., 433:2 (2016), 1243–1260.
[11] A.E. Kovtanyuk, A.Yu. Chebotarev, N.D. Botkin, K.-H. Hoffmann, “Optimal boundary control of a steady-state heat transfer model accounting for radiative effects”, J. Math. Anal. Appl., 439 (2016), 678–689.
[12] Alexander Yu. Chebotarev, Andrey E. Kovtanyuk, Gleb V. Grenkin, Nikolai D. Botkin, and Karl Heinz Hoffmann, “Nondegeneracy of optimality conditions in control problems for a radiative-conductive heat transfer model”, Applied Mathematics and Computation, 289:10 (2016), 371–380.
[13] A.Yu. Chebotarev, G.V. Grenkin, A.E. Kovtanyuk, “Inhomogeneous steady-state problem of complex heat transfer”, ESAIM Math. Model. Numer. Anal., 51:6 (2017), 2511–2519.
[14] A.Yu. Chebotarev, G.V. Grenkin, A.E. Kovtanyuk, N.D. Botkin, K.-H. Hoffmann, “Inverse problem with finite overdetermination for steady-state equations of radiative heat exchange”, J. Math. Anal. Appl., 460:2 (2018), 737–744.
[15] A.Yu. Chebotarev, R. Pinnau, “An inverse problem for a quasi-static approximate model of radiative heat transfer”, J. Math. Anal. Appl., 472:1 (2019), 737–744.
[16] G.V. Grenkin, A.Iu. Chebotarev, “Obratnaia zadacha dlia uravnenii slozhnogo teploobmena” , Zh. vychisl. matem. i matem. fiz., 59:8 (2019), 1420–1430.
[17] Alexander Yu. Chebotarev and Andrey E. Kovtanyuk and Nikolai D. Botkin, “Problem of radiation heat exchange with boundary conditions of the Cauchy type”, Communications in Nonlinear Science and Numerical Simulation, 75 (2019), 262–269.
[18] A.G. Kolobov, T. V. Pak, A. Iu. Chebotarev, “Statsionarnaia zadacha radiatsionnogo teploobmena s granichnymi usloviiami tipa Koshi” , Zh. vychisl. matem. i matem. fiz., 59:7 (2019), 1258–1263.
[19] A.Iu. Chebotarev, “Neodnorodnaia kraevaia zadacha dlia uravnenii slozhnogo teploobmena s frenelevskimi usloviiami sopriazheniia” , Differentsial'nye uravneniia, 56:12 (2020), 1660–1665.
[20] A.Iu. Chebotarev, “Obratnaia zadacha dlia uravnenii slozhnogo teploobmena s frenelevskimi usloviiami sopriazheniia” , Zh. vychisl. matem. i matem. fiz., 61:2 (2021), 303–311.
[21] A.A. Amosov, “Stationary nonlinear nonlocal problem of radiative–conductive heat transfer in a system of opaque bodies with properties depending on the radiation frequency”, Journal of Mathematical Sciences, 164 (2010), 309–344.
[22] A.A. Amosov, “Unique Solvability of Stationary Radiative-Conductive Heat Transfer Problem in a System of Semitransparent Bodies”, Journal of Mathematical Sciences, 224:5 (2017), 618–646.
[23] A.A. Amosov, N. E. Krymov, “On a Nonstandard Boundary Value Problem Arising in Homogenization of Complex Heat Transfer Problems”, Journal of Mathematical Sciences, 244:3 (2020), 357–377.
[24] A. V. Fursikov, Optimal'noe upravlenie raspredelennymi sistemami. Teoriia i prilozheniia, Nauchnaia kniga, 1999.

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