# On some optimal control problems for the heat radiative transfer equation

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cocv146.dvi ESAIM: Control, Optimisation and Calculus of Variations August 2000, Vol. 5, 425{444 URL: http://www.emath.fr/cocv/ ON SOME OPTIMAL CONTROL PROBLEMS FOR THE HEAT RADIATIVE TRANSFER EQUATION � Sandro Manservisi1; 2 and Knut Heusermann2 Abstract. This paper is concerned with some optimal control problems for the Stefan-Boltzmann radiative transfer equation. The objective of the optimisation is to obtain a desired temperature pro�le on part of the domain by controlling the source or the shape of the domain. We present two problems with the same objective functional: an optimal control problem for the intensity and the position of the heat sources and an optimal shape design problem where the top surface is sought as control. The problems are analysed and �rst order necessity conditions in form of variation inequalities are obtained. AMS Subject Classi�cation. 49N50, 80A23. Received August 3, 1999. Revised June 2, 2000. 1. Introduction Radiative heat exchange can be found in many natural and engineering processes. At low temperature this exchange is small but at high temperature could be the leading form of heat transfer and in many situations, like in some forming processes, it is appropriate to model the heat exchange with the pure radiation equation. Motivated by the desire to remove hot spots and control the temperature distribution we present a systematic mathematical theory for the optimisation of systems which are appropriately described by this model. We consider radiative heat exchange in a two-dimensional domain Ω with boundary Γ and radiative source f governed by the integral boundary equations [16] u(s) = �T 4(s) s 2 Γ u(s)− R Γ K(s; s0)u(s0) ds0 = f(s) s 2 Γ; (1.1) where T is the temperature, u is the radiosity, � is the Stefan-Boltzmann constant and K(s; s0) is the kernel representing the radiative heat exchange between s and s0 [16]. Keywords and phrases: Optimal control, heat radiative transfer, optimal shape design. � S.M. was supported by European c

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