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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 $$]0,t_{f}[$$ ] 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...
The problem of determining the source term F(x, t) in the linear parabolic equation ut = (k(x)ux(x, t))x + F(x, t) from the measured data at the final time u(x, T) = µ(x) is formulated. It is proved that the Fréchet derivative of the cost functional J(F) = ‖µT (x) − u(x, T)‖ 02 can be formulated via the solution of the adjoint parabolic problem. Lipschitz continuity of the gradient is...
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