Computational Optimization and Applications

, Volume 44, Issue 2, pp 183–212 | Cite as

Regularization of state-constrained elliptic optimal control problems with nonlocal radiation interface conditions

  • C. Meyer
  • I. YouseptEmail author


A state-constrained optimal control problem with nonlocal radiation interface conditions arising from the modeling of crystal growth processes is considered. The problem is approximated by a Moreau-Yosida type regularization. Optimality conditions for the regularized problem are derived and the convergence of the regularized problems is shown. In the last part of the paper, some numerical results are presented.


Nonlinear optimal control Nonlocal radiation interface conditions State constraints First-order necessary conditions Second-order sufficient conditions Moreau-Yosida approximation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adams, R.A.: Sobolev Spaces. Academic Press, San Diego (1978) Google Scholar
  2. 2.
    Alibert, J.-J., Raymond, J.-P.: Boundary control of semilinear elliptic equations with discontinuous leading coefficients and unbounded controls. Numer. Funct. Anal. Optim. 3&4, 235–250 (1997) CrossRefMathSciNetGoogle Scholar
  3. 3.
    Atkinson, K., Chandler, G.: The collocation method for solving the radiosity equation for unoccluded surfaces. J. Integral Equ. Appl. 10, 253–290 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bergounioux, M., Kunisch, K.: Primal-dual active set strategy for state-constrained optimal control problems. Comput. Optim. Appl. 22, 193–224 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bergounioux, M., Ito, K., Kunisch, K.: Primal-dual strategy for constrained optimal control problems. SIAM J. Control Optim. 37, 1176–1194 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Casas, E.: Control of an elliptic problem with pointwise state constraints. SIAM J. Control Optim. 4, 1309–1322 (1986) CrossRefMathSciNetGoogle Scholar
  7. 7.
    Casas, E.: Boundary control of semilinear elliptic equations with pointwise state constraints. SIAM J. Control Optim. 31, 993–1006 (1993) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Casas, E., Tröltzsch, F.: Error estimates for the finite-element approximation of a semilinear elliptic control problem. Control Cybern. 31, 695–712 (2002) zbMATHGoogle Scholar
  9. 9.
    Casas, E., De Los Reyes, J.C., Tröltzsch, F.: Sufficient second order optimality conditions for semilinear control problems with pointwise state constraints (2007, submitted) Google Scholar
  10. 10.
    de los Reyes, J., Kunisch, K.: A semismooth Newton method for control constrained boundary optimal control of the Navier-Stokes equations. Nonlinear Anal. 62, 1289–1316 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Elschner, J., Rehberg, J., Schmidt, G.: Optimal regularity for elliptic transmission problems including C 1 interfaces. Interfaces Free Bound. 2, 233–252 (2007) MathSciNetGoogle Scholar
  12. 12.
    Gröger, K.: A W 1,p-estimate for solutions to mixed boundary value problems for second order elliptic differential equations. Math. Ann. 283, 679–687 (1989) zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Hintermüller, M., Kunisch, K.: Feasible and non-interior path-following in constrained minimization with low multiplier regularity. Report 01-05, Department of Mathematics and Scientific Computing, University of Graz, Heinrichstraße 36, A-8010 Graz, Austria, October 2005 Google Scholar
  14. 14.
    Hintermüller, M., Ito, K., Kunisch, K.: The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim. 13, 865–888 (2003) zbMATHCrossRefGoogle Scholar
  15. 15.
    Ito, K., Kunisch, K.: Augmented Lagrangian methods for nonsmooth, convex optimization in Hilbert spaces. Nonlinear Anal. TMA 41, 591–616 (2000) CrossRefMathSciNetGoogle Scholar
  16. 16.
    Ito, K., Kunisch, K.: Semi-smooth Newton methods for state-constrained optimal control problems. Syst. Control. Lett. 50, 221–228 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Ito, K., Kunisch, K.: The primal-dual active set method for nonlinear optimal control problems with bilateral constraints. SIAM J. Control Optim. 43, 357–376 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Klein, O., Philip, P., Sprekels, J.: Modeling and simulation of sublimation growth of sic bulk single crystals. Interfaces Free Bound. 6, 295–314 (2004) zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Konstantinov, A.: Sublimation growth of SiC. In: Harris, G. (ed.) Properties of Silicon Carbide. EMIS Datareview Series, pp. 170–203. Institution of Electrical Engineers, INSPEC, London (1995). Chap. 8.2 Google Scholar
  20. 20.
    Laitinen, M., Tiihonen, T.: Conductive-radiative heat transfer in grey materials. Q. Appl. Math. 59, 737–768 (2001) zbMATHMathSciNetGoogle Scholar
  21. 21.
    Meyer, C.: Optimal control of semilinear elliptic equations with applications to sublimation crystal growth. Ph.D. Thesis, TU-Berlin (2006) Google Scholar
  22. 22.
    Meyer, C., Philip, P., Tröltzsch, F.: Optimal control of a semilinear PDE with nonlocal radiation interface conditions. SIAM J. Control Optim. 45, 699–721 (2006) CrossRefMathSciNetGoogle Scholar
  23. 23.
    Meyer, C., Yousept, I.: State-constrained optimal control of semilinear elliptic equations with nonlocal radiation interface conditions (2007, submitted) Google Scholar
  24. 24.
    Philip, P.: Transient numerical simulation of sublimation growth of SiC bulk single crystals. Modeling, finite volume method, results. Ph.D. Thesis, Department of Mathematics, Humboldt University of Berlin (2003) Google Scholar
  25. 25.
    Rost, H.-J., Siche, D., Dolle, J., Eiserbeck, W., Müller, T., Schulz, D., Wagner, G., Wollweber, J.: Influence of different growth parameters and related conditions on 6H-SiC crystals grown by the modified Lely method. Mater. Sci. Eng. 61–62, 68–72 (1999) CrossRefGoogle Scholar
  26. 26.
    Tiihonen, T.: A nonlocal problem arising from heat radiation on non-convex surfaces. Eur. J. App. Math. 8, 403–416 (1997) zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Weierstrass Institute for Applied Analysis and StochasticsBerlinGermany
  2. 2.Institut für MathematikTechnische Universität BerlinBerlinGermany

Personalised recommendations