Far Eastern Mathematical Journal

To content of the issue


An algorithm for solving the problem of boundary optimal control in a complex heat transfer model


Grenkin G.V.

2016, issue 1, Ñ. 24-38


Abstract
A nonstationary model of complex heat transfer, which includes $P_1$ approximation for the radiative heat transfer equation, is considered. The optimal control problem consists in determination of the boundary reflection coefficient within the specified range in order to minimize the cost functional. The considered algorithm for solving the control problem is based on the fact, that the optimal control satisfies the bang-bang principle, and employs the idea of the gradient descent method.

Keywords:
radiative heat transfer, diffusion approximation, optimal control, bang-bang, gradient descent method

Download the article (PDF-file)

References

[1] D. Clever, J. Lang, D. Schroder, «Model hierarchy-based optimal control of radiative heat transfer», Int. J. Comput. Sci. Eng., 9:5–6 (2014), 509–525.
[2] A. Farina, A. Klar, R. M. M. Mattheij, A. Mikelic, N. Siedow, Mathematical models in the manufacturing of glass, Lecture Notes in Mathematics, Springer, 2011.
[3] G. Thommes, R. Pinnau, M. Sead, T. Gotz, A. Klar, «Numerical methods and optimal control for glass cooling processes», Transport Theory Statist. Phys., 31:4–6 (2002), 513–529.
[4] R. Pinnau, G. Thommes, «Optimal boundary control of glass cooling processes», Math. Methods Appl. Sci., 27:11 (2004), 1261–1281.
[5] R. Pinnau, «Analysis of optimal boundary control for radiative heat transfer modeled by the SP1-system», Commun. Math. Sci., 5:4 (2007), 951–969.
[6] R. Pinnau, A. Schulze, «Newton's method for optimal temperature-tracking of glass cooling processes», Inverse Probl. Sci. Eng., 15:4 (2007), 303–323.
[7] M. Frank, A. Klar, R. Pinnau, «Optimal control of glass cooling using simplified PN theory», Transport Theory Statist. Phys., 39:2–4 (2010), 282–311.
[8] D. Clever, J. Lang, «Optimal control of radiative heat transfer in glass cooling with restrictions on the temperature gradient», Optimal Control Appl. Methods, 33:2 (2012), 157–175.
[9] O. Tse, R. Pinnau, «Optimal control of a simplified natural convection-radiation model», Commun. Math. Sci., 11:3 (2013), 79–707.
[10] 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., 439:2 (2016), 678–689.
[11] A.E. Kovtanyuk, A.Yu. Chebotarev, N.D. Botkin, K.-H. Hoffmann, «Theoretical analysis of an optimal control problem of conductive-convective-radiative heat transfer», J. Math. Anal. Appl., 412:1 (2014), 520–528.
[12] G.V. Grenkin, «Optimal'noe upravlenie v nestacionarnoj zadache slozhnogo teploobmena», Dal'nevost. matem. zhurn., 14:2 (2014), 160–172.
[13] G.V. Grenkin, A.Yu. Chebotarev, A.E. Kovtanyuk, N.D. Botkin, K.-H. Hoffmann, «Boundary optimal control problem of complex heat transfer model», J. Math. Anal. Appl., 433:2 (2016), 1243–1260.
[14] A. Munch, F. Periago, «Numerical approximation of bang-bang controls for the heat equation: An optimal design approach», Systems Control Lett., 62:8 (2013), 643–655.
[15] K. Glashoff, E. Sachs, «On theoretical and numerical aspects of the bang-bang-principle», Numer. Math., 29:1 (1977), 93–113.
[16] K. Deckelnick, M. Hinze, «A note on the approximation of elliptic control problems with bang-bang controls», Comput. Optim. Appl., 51:2 (2012), 931–939.
[17] C.Y. Kaya, S.K. Lucas, S.T. Simakov, «Computations for bang-bang constrained optimal control using a mathematical programming formulation», Optimal Control Appl. Methods, 25:6 (2004), 295–308.
[18] T. Taleshian, A. Ranjbar Noei, R. Ghaderi, «IPSO-SQP algorithm for solving time optimal bang-bang control problems and its application on autonomous underwater vehicle», Journal of Advances in Computer Research, 5:2 (2014), 69–88.
[19] H. Maurer, H.D. Mittelmann, «Optimization techniques for solving elliptic control problems with control and state constraints: Part 1. Boundary control», Comput. Optim. Appl., 16:1 (2000), 29–55.
[20] H. Maurer, H.D. Mittelmann, «Optimization techniques for solving elliptic control problems with control and state constraints: Part 2. Distributed control», Comput. Optim. Appl., 18:2 (2001), 141–160.
[21] A. Borzi, K. Kunisch, «A multigrid scheme for elliptic constrained optimal control problems», Comput. Optim. Appl., 31:3 (2005), 309–333.
[22] M.J. Zandvliet, O.H. Bosgra, J.D. Jansen, P.M.J. Van den Hof, J.F.B.M. Kraaijevanger, «Bang-bang control and singular arcs in reservoir flooding», J. Pet. Sci. Eng., 58:1–2 (2007), 186–200.
[23] N.N. Kalitkin, Chislennye metody, Nauka, M., 1978.
[24] R. Barrett, M. Berry, T.F. Chan, J. Demmel, J.M. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, H. Van der Vorst, Templates for the solution of linear systems: building blocks for iterative methods, 2nd Edition, SIAM Press, Philadelphia, 1994.
[25] The Iterative Template Library http://www.osl.iu.edu/research/itl/.
[26] G.V. Grenkin, A.Ju. Chebotarev, “Optimal'noe upravlenie granichnym kojefficientom v polustacionarnoj modeli slozhnogo teploobmena v trehmernoj oblasti”, Svidetel'stvo o gosudarstvennoj registracii programmy dlja JeVM 2016611950, 15.02.2016, Zaregistrirovano v Reestre programm dlja JeVM.

To content of the issue