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Control of the radiative heating of a semi-transparent body

  • Hawraa Nabolsi
  • Luc Paquet
  • Ali Wehbe
Article
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Abstract

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.

Keywords

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 

Notes

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Copyright information

© Springer Nature Switzerland AG 2018

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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