Control of the radiative heating of a glass plate
Résumé
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(x, t) at time t along the flat glass thickness x during the fixed time of heating ] 0 , tf[ , 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⟼ u(t) to be an optimal control. The state space and the constraining mapping are carefully defined in order for the constraining mapping (T, u) ↦ e(T, u) to be Fréchet differentiable and its derivative with respect to the temperature T at a point (T, u) to be invertible. This, allows us to apply the implicit function theorem in order to compute the derivative of the reduced cost functional J^ (u) at a control u∈ H1(] 0 , tf[).