Control of the radiative heating of a semi-transparent body
Résumé
In a preceding paper, we have studied the radiative heating of a semi-transparent body Ω (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 tf, 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)↦T(x,t) inside the semi-transparent body Ω near a desired temperature distribution Td(⋅,⋅) during the time interval of radiative heating ]0,tf[ by acting on u, the purpose being to deform Ω to manufacture a new object. In this respect, we introduce the appropriate cost functional and the set of admissible controls Uad, 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↦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.