×

On some optimal control problems for the heat radiative transfer equation. (English) Zbl 0952.49035

Summary: 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 necessary conditions in form of variational inequalities are obtained.

MSC:

49Q10 Optimization of shapes other than minimal surfaces
49N45 Inverse problems in optimal control
80A23 Inverse problems in thermodynamics and heat transfer
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] F. Abergel and R. Temam, On some control problems in fluid mechanics. Theoret. Computational Fluid Dynamics 1 ( 1990) 303-326. Zbl0708.76106 · Zbl 0708.76106 · doi:10.1007/BF00271794
[2] R. Adams, Sobolev Spaces. Academic Press, New York ( 1975). Zbl0314.46030 MR450957 · Zbl 0314.46030
[3] V. Alekseev, V. Tikhomirov and S. Fomin, Optimal Control. Consultants Bureau, New York ( 1987). Zbl0689.49001 MR924574 · Zbl 0689.49001
[4] I. Babuska, The finite element method with Lagrangian multipliers. Numer. Math. 16 ( 1973) 179-192. Zbl0258.65108 MR359352 · Zbl 0258.65108 · doi:10.1007/BF01436561
[5] D.M. Bedivan and G.J. Fix, An extension theorem for the space Hdiv. Appl. Math. Lett. (to appear). Zbl0915.46024 · Zbl 0915.46024 · doi:10.1016/0893-9659(96)00066-3
[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). Zbl0672.65001 MR1015713 · Zbl 0672.65001
[8] P. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam ( 1978). Zbl0383.65058 MR520174 · Zbl 0383.65058
[9] J.E. Dennis and R.B. Schnabel, Numerical methods for unconstrained optimisation and non-linear equations. Prentice-Hall Inc., New Jersey ( 1983). Zbl0579.65058 · Zbl 0579.65058
[10] V. Girault and P. Raviart, The Finite Element Method for Navier-Stokes Equations: Theory and Algorithms. Springer-Verlag, New York ( 1986). Zbl0585.65077 MR851383 · Zbl 0585.65077
[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). Zbl0963.35150 MR1759904 · Zbl 0963.35150 · doi:10.1137/S0036142997329414
[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). Zbl0938.35118 MR1720145 · Zbl 0938.35118 · doi:10.1137/S0363012998337400
[13] J. Haslinger and P. Neittaanmäki, Finite Element Approximation for Optimal Shape Design. Wiley, Chichester ( 1988). Zbl0713.73062 MR982710 · Zbl 0713.73062
[14] K. Heusermann and S. Manservisi, Optimal design for heat radiative transfer systems. Comput. Methods Appl. Mech. Engrg. (to appear). · Zbl 0952.49035
[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] O. Pironneau, On optimal design in fluid mechanics. J. Fluid. Mech. 64 ( 1974) 97-110. Zbl0281.76020 MR347229 · Zbl 0281.76020 · doi:10.1017/S0022112074002023
[19] O. Pironneau, Optimal shape design for elliptic systems. Springer, Berlin ( 1984). Zbl0534.49001 MR725856 · Zbl 0534.49001
[20] R.E. Showalter, Hilbert Space Methods for Partial Differential Equations. Electron. J. Differential Equations ( 1994) http://ejde.math.swt.edu/mono-toc.html Zbl0991.35001 MR1302484 · Zbl 0991.35001
[21] J. Sokolowski and J. Zolesio, Introduction to shape optimisation: Shape sensitivity analysis. Springer, Berlin ( 1992). Zbl0761.73003 · Zbl 0761.73003
[22] T. Tiihonen, Stefan-Boltzmann radiation on Non-convex Surfaces. Math. Methods Appl. Sci. 20 ( 1997) 47-57. Zbl0872.35044 MR1429330 · Zbl 0872.35044 · doi:10.1002/(SICI)1099-1476(19970110)20:1<47::AID-MMA847>3.0.CO;2-B
[23] T. Tiihonen, Finite Element Approximations for a Beat 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). Zbl0595.49001 MR866483 · Zbl 0595.49001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.