Advertisement

Afrika Matematika

, Volume 27, Issue 3–4, pp 673–699 | Cite as

Control of the radiative heating of a glass plate

  • Luc Paquet
Article

Abstract

In a preceding paper (MSIA, 2012), we have studied the radiative heating of an infinite horizontal glass plate. Here, we want to control the temperature T(xt) at time t along the flat glass thickness x during the fixed time of heating \(]0,t_{f}[\), by acting on the temperature u(t) of the black radiative source S, placed above the glass plate. A first order necessary condition in the form of a variational inequality is derived for a control \(u:t\longmapsto u(t)\) to be an optimal control. The state space and the constraining mapping are carefully defined in order for the constraining mapping \(\left( T,u\right) \mapsto e\left( T,u\right) \) to be Fréchet differentiable and its derivative with respect to the temperature T at a point \(\left( T,u\right) \) to be invertible. This, allows us to apply the implicit function theorem in order to compute the derivative of the reduced cost functional \(\hat{J}(u)\) at a control \(u\in H^{1}(]0,t_{f}[)\).

Keywords

Planck function Radiative transfer equation in a glass plate Nonlinear heat-conduction equation Cost functional Reduced cost functional Fréchet derivative Implicit function theorem Adjoint problem Gradient of the reduced cost functional 

Mathematics Subject Classification

35K20 35K55 45K05 49J20 49K20 

Supplementary material

13370_2015_371_MOESM1_ESM.tex (15 kb)
Supplementary material 1 (tex 15 KB)
13370_2015_371_MOESM2_ESM.tex (3 kb)
Supplementary material 2 (tex 3 KB)

References

  1. 1.
    Bonnans, F.: Optimisation Continue Cours et problèmes corrigés. Dunod, Paris (2006)Google Scholar
  2. 2.
    Brézis, H.: Analyse fonctionnelle Théorie et applications. Masson, Paris (Zbl 1147.46300) (1993)Google Scholar
  3. 3.
    Casas, E.: Pontryagin’s principle for state-constrained boundary control problems of semilinear parabolic equations. SIAM J. Control Optim. 35(4), 1297–1327 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Clever, D., Lang, J.: Optimal control of radiative heat transfer in glass cooling with restrictions on the temperature gradient. Optim. Control Appl. Meth. (2011) (Published online in Wiley Online Library, 1002). doi: 10.1002/oca.984
  5. 5.
    Dautray, R., Lions, J.-L.: Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques Volume 8 Evolution: semi-groupe, variationnel. Masson, Paris (Zbl 0749.35004) (1988)Google Scholar
  6. 6.
    Desvignes, F.: Rayonnements optiques Radiométrie-Photométrie. Masson, Paris (1991)Google Scholar
  7. 7.
    Dieudonné, J.: Foundations of Modern Analysis Enlarged and Corrected Printing. Academic Press, New York (1969)zbMATHGoogle Scholar
  8. 8.
    Héron, B., Issard-Roch, F., Picard, C.: Analyse numérique Exercices et problèmes corrigés. Dunod, Paris (1999)Google Scholar
  9. 9.
    Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints. Springer, Berlin (Zbl 1167.49001) (2009)Google Scholar
  10. 10.
    Hottel, H.C., Sarofim, A.F.: Radiative Transfer. McGraw-Hill, New York (1967)Google Scholar
  11. 11.
    Lang, J.: Adaptive computation for boundary control of radiative heat tranfer in glass. J. Comput. Appl. Math. 183, 312–326 (Zbl 1100.65073) (2005)Google Scholar
  12. 12.
    Lions, J.-L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Paris (Zbl 0189.40603) (1969)Google Scholar
  13. 13.
    Modest, M.F.: Radiative Heat Transfer, 2nd edn. Academic Press, New York (2003)zbMATHGoogle Scholar
  14. 14.
    Paquet, L., El Cheikh, R., Lochegnies, D., Siedow, N.: Radiative heating of a glass plate. Math. Action 5, 1–30 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Pinnau, R.: Analysis of optimal boundary control for radiative heat ransfer modeled by the SP\(_{1}\) -system. Commun. Math. Sci. 54:951–969 ( Zbl 1145.49015) (2007)Google Scholar
  16. 16.
    Planck, M.: The Theory of Heat Radiation. Dover, New York (Zbl 0127.21701) (1991)Google Scholar
  17. 17.
    Raymond, J.P., Zidani, H.: Hamiltonian Pontryagin’s Principles for Control Problems Governed by Semilinear Parabolic Equations. Appl. Math. Optim. 39, 143–177 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Rudin, W.: Real and Complex Analysis. Series in Higher Mathematics, 2nd edn. McGraw-Hill, Nre York (1974)zbMATHGoogle Scholar
  19. 19.
    Siedow, N., Grosan, T., Lochegnies, D., Romero, E.: Application of a new method for radiative heat tranfer to flat glass tempering. J. Am. Ceram. Soc. 88(8), 2181–2187 (2005)CrossRefGoogle Scholar
  20. 20.
    Soudre, L.: “Etude numérique et expérimentale du thermoformage d’une plaque de verre”, thèse soutenue le 9 décembre 2008, Nancy-Université Henri Poincaré, laboratoire LEMTA, tel-00382905, version 1–11 May 2009Google Scholar
  21. 21.
    Taine, J., Iacona, E., Petit, J.-P.: Transferts Thermiques Introduction aux transferts d’énergie, 4th edn. Dunod, Paris (2008)Google Scholar
  22. 22.
    Taine, J., Petit, J.-P.: Heat Transfer. Prentice Hall, Englewood Cliffs (Zbl 0818.73001) (1993)Google Scholar
  23. 23.
    Treves, F.: Basic Linear Partial Differential Equations. Academic Press, New York (1975)zbMATHGoogle Scholar
  24. 24.
    Tröltzsch, F.: Optimal Control of Partial Differential Equations Theory, Methods and Applications. Translated from the German text Optimale Steuerung partieller Differentialgleichungen (2005) by Jürgen Sprekels, GSM, vol. 112. Applied Mathematics, AMS Providence, Rhode Island (2010)Google Scholar
  25. 25.
    Zeidler, E.: Nonlinear Functional Analysis and its Applications II/A: Linear Monotone Operators. Springer, Berlin (1990)CrossRefzbMATHGoogle Scholar

Copyright information

© African Mathematical Union and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.LAMAV-EDP, FR (Fédération de Recherches) no 2956, Institut des Sciences et Techniques de Valenciennes (ISTV2)Université de Valenciennes et du Hainaut-Cambrésis (UVHC)Valenciennes Cedex 9France

Personalised recommendations