Control of the radiative heating of a semi-transparent body

  • Hawraa Nabolsi
  • Luc PaquetEmail author
  • Ali Wehbe


In a preceding paper, we have studied the radiative heating of a semi-transparent body \(\varOmega \) (e.g., glass) by a black radiative source S surrounding it, black source at absolute uniform temperature u(t) at time t between time 0 and time \(t_\mathrm{f}\), the final time of the radiative heating. This problem has been modeled by an appropriate coupling between quasi-steady radiative transfer boundary value problems with nonhomogeneous reflectivity boundary conditions (one for each wavelength band in the semi-transparent electromagnetic spectrum of the glass) and a nonlinear heat conduction evolution equation with a nonlinear Robin boundary condition which takes into account those wavelengths for which the glass behaves like an opaque body. In the present paper, u being considered as the control variable, we want to adjust the absolute temperature distribution \((x,t) \mapsto T(x,t)\) inside the semi-transparent body \(\varOmega \) near a desired temperature distribution \(T_\mathrm{d}(\cdot ,\cdot )\) during the time interval of radiative heating \(]0,t_\mathrm{f}[\) by acting on u, the purpose being to deform \(\varOmega \) to manufacture a new object. In this respect, we introduce the appropriate cost functional and the set of admissible controls \(U_\mathrm{ad}\), for which we prove the existence of optimal controls. Introducing the state space and the state equation, a first-order necessary condition for a control \(u:t \mapsto u(t)\) to be optimal is then derived in the form of a Variational Inequality by using the implicit function theorem and the adjoint problem. We close this paper by some numerical considerations.


Optimal control problem Nonlinear parabolic equation with an integral 0-order term Nonlinear boundary condition of the Robin type Cost functional First-order necessary optimality condition Backward parabolic linear equation 

Mathematics Subject Classification

49J20 35K61 35Q79 80M50 



  1. 1.
    Agboka, K., Béchet, F., Siedow, N., Lochegnies, D.: Influence of radiative heat transfer model on the computation of residual stresses in glass tempering process. Int. J. Appl. Glass Sci. 9, 1–17 (2017)Google Scholar
  2. 2.
    Bergounioux, M.: Optimisation et contrôle des systèmes linéaires Cours et Exercices Avec Solutions. Dunod, Paris (2001)Google Scholar
  3. 3.
    Boubaker, M.B., Le Corre, B., Meshaka, Y., Jeandel, G.: Finite element simulation of the slumping process of a glass plate using 3-D generalized viscoelastic Maxwell model. J. Non-Cryst. Solids 405, 45–54 (2014)CrossRefGoogle Scholar
  4. 4.
    Brézis, H.: Analyse Fonctionnelle Théorie et Applications. Masson, Paris (1993)zbMATHGoogle Scholar
  5. 5.
    Bonnans, F.: Optimisation Continue Cours et Problèmes Corrigés. Dunod, Paris (2006)Google Scholar
  6. 6.
    Casas, E.: Pontryagin’s principle for state-constrained boundary control problems of semilinear parabolic equations. SIAM J. Control Optim. 35(4), 1297–1327 (1997)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Clever, D., Lang, J.: Optimal control of radiative heat transfer in glass cooling with restrictions on the temperature gradient. Optim. Control Appl. Methods 33, 157–175 (2012)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Dautray, R., Lions, J.-L.: Mathematical Analysis and Numerical Methods for Sciences and Technology, Functional and Variational Methods, vol. 2. Springer, Berlin (2000)zbMATHGoogle Scholar
  9. 9.
    Dautray, R., Lions, J.-L.: Mathematical Analysis and Numerical Methods for Sciences and Technology, Evolution Problems I, vol. 5. Springer, Berlin (2000)zbMATHGoogle Scholar
  10. 10.
    Dieudonné, J.: Foundations of Modern Analysis Enlarged and Corrected Printing. Academic Press, New York (1969)zbMATHGoogle Scholar
  11. 11.
    Grisvard, P.: Commutativité de deux foncteurs d’interpolation et applications. J. Math. Pures Appl. 45, 143–290 (1966)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Héron, B., Issard-Roch, F., Picard, C.: Analyse Numérique Exercices et Problèmes Corrigés. Dunod, Paris (1999)Google Scholar
  13. 13.
    Herty, M.: Results on an optimality system in radiative transfer. PAMM Proc. Appl. Math. Mech. 6, 787–788 (2006)CrossRefGoogle Scholar
  14. 14.
    Herty, M., Pinnau, R., Seaïd, M.: Optimal control in radiative transfer. Optim. Methods Softw. 22(6), 917–936 (2007)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints, Mathematical Modelling: Theory and Applications, vol. 23. Springer, Cham (2009)zbMATHGoogle Scholar
  16. 16.
    Lions, J.-L.: Quelques Méthodes de Résolution des Problèmes Aux Limites Non Linéaires. Dunod, Paris (1969)zbMATHGoogle Scholar
  17. 17.
    Modest, M.F.: Radiative Heat Transfer, 2nd edn. Academic Press, Cambridge (2003)zbMATHGoogle Scholar
  18. 18.
    Nabolsi, H.: Contrôle Optimal des Equations d’Evolution et ses Applications. Ph.D. thesis (written in English), University of Valenciennes and Hainaut-Cambrésis, Laboratory LAMAV. Accessed 17 July 2018
  19. 19.
    Paquet, L., Nabolsi, H.: The radiative transfer equation with the reflectivity boundary condition coupled with the heat conduction equation. Math. Methods Appl. Sci. 41(13), 5254–5273 (2018)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Paquet, L., El Cheikh, R., Lochegnies, D., Siedow, N.: Radiative heating of a glass plate. Math. Action 5, 1–30 (2012)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Paquet, L.: Control of the Radiative Heating of a Glass Plate: Theory and Discretization, report of the LAMAV laboratory (2015), Université de Valenciennes et du Hainaut-Cambrésis, FranceGoogle Scholar
  22. 22.
    Paquet, L.: Control of the radiative heating of a glass plate. Afr. Mat. 27(3), 673–699 (2016)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Parsa, M.H., Rad, M., Shahhosseini, M.R., Shahhosseini, M.H.: Simulation of windscreens bending using viscoplastic formulation. J. Mater. Process. Technol. 170, 298–303 (2005)CrossRefGoogle Scholar
  24. 24.
    Pinnau, R.: Analysis of optimal boundary control for radiative heat ransfer modeled by the SP1-system. Commun. Math. Sci. 5(4), 951–969 (2007)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Showalter, R.E.: Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Mathematical Surveys and Monographs, vol. 49. American Mathematical Society, Providence (1997)Google Scholar
  26. 26.
    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 (2009)Google Scholar
  27. 27.
    Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators, North-Holland Mathematical Library, vol. 18. North-Holland Publishing Company, Amsterdam (1978)Google Scholar
  28. 28.
    Tröltzsch, F.: Optimal Control of Partial Differential Equations Theory, Methods and Applications, Translated by Jürgen Sprekels from the German text “Optimale Steuerung partieller Differentialgleichungen” (2005), Graduate Studies in Mathematics (GSM), Vol. 112, Applied Mathematics, American Mathematical Society (AMS) Providence, Rhode Island (2010)Google Scholar

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Authors and Affiliations

  1. 1.EA 4015-LAMAV, FR CNRS 2956Université Polytechnique Hauts-De-FranceValenciennesFrance
  2. 2.Faculté des Sciences IUniversité LibanaiseHadeth, BeirutLebanon

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