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On some optimal control problems for the heat radiative transfer equation

  • Sandro Manservisi (a1) (a2) and Knut Heusermann (a2)


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 profile 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 first order necessity conditions in form of variation inequalities are obtained.



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[1] Abergel, F. and Temam, R., On some control problems in fluid mechanics. Theoret. Computational Fluid Dynamics 1 (1990) 303-326.
[2] R. Adams, Sobolev Spaces. Academic Press, New York (1975).
[3] V. Alekseev, V. Tikhomirov and S. Fomin, Optimal Control. Consultants Bureau, New York (1987).
[4] Babuska, I., The finite element method with Lagrangian multipliers. Numer. Math. 16 (1973) 179-192.
[5] D.M. Bedivan and G.J. Fix, An extension theorem for the space H div. Appl. Math. Lett. (to appear).
[6] N. Di Cesare, O. Pironneau and E. Polak, Consistent approximations for an optimal design problem. Report 98005 Labotatoire d'analyse numérique, Paris, France (1998).
[7] P. Ciarlet, Introduction to Numerical Linear Algebra and Optimization. Cambridge University, Cambridge (1989).
[8] P. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978).
[9] J.E. Dennis and R.B. Schnabel, Numerical methods for unconstrained optimisation and non-linear equations. Prentice-Hall Inc., New Jersey (1983).
[10] V. Girault and P. Raviart, The Finite Element Method for Navier-Stokes Equations: Theory and Algorithms. Springer-Verlag, New York (1986).
[11] M. Gunzburger and S. Manservisi, Analysis and approximation of the velocity tracking problem for Navier-Stokes flows with distributed control. SIAM J. Numer. Anal. (to appear).
[12] M. Gunzburger and S. Manservisi, The velocity tracking problem for for Navier-Stokes flows with bounded distributed control. SIAM J. Control Optim. (to appear).
[13] J. Haslinger and P. Neittaanmäki, Finite Element Approximation for Optimal Shape Design. Wiley, Chichester (1988).
[14] K. Heusermann and S. Manservisi, Optimal design for heat radiative transfer systems. Comput. Methods Appl. Mech. Engrg. (to appear).
[15] F.P. Incropera and D.P. DeWitt, Fundamentals of Heat and Mass Transfer. Wiley, New York (1990).
[16] M. Modest, Radiative heat transfer. McGraw-Hill, New York (1993).
[17] O. Pironneau, Optimal shape design in fluid mechanics. Thesis, University of Paris (1976).
[18] Pironneau, O., On optimal design in fluid mechanics. J. Fluid. Mech. 64 (1974) 97-110.
[19] O. Pironneau, Optimal shape design for elliptic systems. Springer, Berlin (1984).
[20] R.E. Showalter, Hilbert Space Methods for Partial Differential Equations. Electron. J. Differential Equations (1994)
[21] J. Sokolowski and J. Zolesio, Introduction to shape optimisation: Shape sensitivity analysis. Springer, Berlin (1992).
[22] Tiihonen, T., Stefan-Boltzmann radiation on Non-convex Surfaces. Math. Methods Appl. Sci. 20 (1997) 47-57.
[23] T. Tiihonen, Finite Element Approximations for a Heat Radiation Problem. Report 7/1995, Dept. of Mathematics, University of Jyväskylä (1995).
[24] V. Tikhomirov, Fundamental Principles of the Theory of Extremal Problems. Wiley, Chichester (1986).


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On some optimal control problems for the heat radiative transfer equation

  • Sandro Manservisi (a1) (a2) and Knut Heusermann (a2)


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