Afrika Matematika

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

Control of the radiative heating of a glass plate

  • Luc Paquet


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 \(u\in H^{1}(]0,t_{f}[)\).


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)


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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

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