Hostname: page-component-848d4c4894-2pzkn Total loading time: 0 Render date: 2024-05-15T09:35:13.718Z Has data issue: false hasContentIssue false
  • English
  • Français

Second-order sufficient optimality conditions for a semilinear optimal control problem with nonlocal radiation interface conditions

Published online by Cambridge University Press:  20 July 2007

Christian Meyer
Christian Meyer*
Affiliation:
WIAS - Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, D-10117 Berlin, Germany; cmeyer@math.tu-berlin.de
Get access

Abstract

We consider a control constrained optimal control problem governed by a semilinear elliptic equation with nonlocal interface conditions. These conditions occur during the modeling of diffuse-gray conductive-radiative heat transfer. After stating first-order necessary conditions, second-order sufficient conditions are derived that account for strongly active sets. These conditions ensure local optimality in an Ls-neighborhood of a reference function whereby the underlying analysis allows to use weaker norms than $L^\infty$.

Keywords

Optimal controlsemilinear elliptic equationsnonlocal interface conditionssecond-order sufficient optimality conditions
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 13 , Issue 4 , October 2007 , pp. 750 - 775
Copyright
© EDP Sciences, SMAI, 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bonnans, J., Second order analysis for control constrained optimal control problems of semilinear elliptic systems. Appl. Math. Optim. 38 (1998) 303–325. CrossRef
H. Brezis, Analyse fonctionelle. Masson, Paris (1983).
Casas, E. and Mateos, M., Second order sufficient optimality conditions for semilinear elliptic control problems with finitely many state constraints. SIAM J. Control Optim. 40 (2002) 1431–1454. CrossRef
Casas, E., Tröltzsch, F. and Unger, A., Second order sufficient optimality conditions for a nonlinear elliptic control problem. J. Anal. Appl. 15 (1996) 687–707.
Dontchev, A.L., Hager, W.W., Poore, A.B. and Yang, B., Optimality, stability, and convergence in optimal control. Appl. Math. Optim. 31 (1995) 297–326.
Klein, O., Philip, P. and Sprekels, J., Modeling and simulation of sublimation growth of SiC bulk single crystals. Interfaces Free Boundaries 6 (2004) 295–314. CrossRef
Laitinen, M. and Tiihonen, T., Conductive-radiative heat transfer in grey materials. Quart. Appl. Math. 59 (2001) 737–768. CrossRef
C. Meyer, P. Philip, and F. Tröltzsch, Optimal control of a semilinear PDE with nonlocal radiation interface conditions. SIAM J. Control Optim. 45 (2006) 699–721.
Rost, H.-J., Siche, D., Dolle, J., Eiserbeck, W., Müller, T., Schulz, D., Wagner, G. and Wollweber, J., Influence of different growth parameters and related conditions on 6H-SiC crystals grown by the modified Lely method. Mater. Sci. Eng. B 61-62 (1999) 68–72. CrossRef
Tiihonen, T., A nonlocal problem arising from heat radiation on non-convex surfaces. Eur. J. App. Math. 8 (1997) 403–416.
Tiihonen, T., Stefan-Boltzmann radiation on non-convex surfaces. Math. Meth. Appl. Sci. 20 (1997) 47–57. 3.0.CO;2-B>CrossRef
Tröltzsch, F. and Wachsmuth, D., Second-order sufficient optimality conditions for the optimal control of Navier-Stokes equations. ESAIM: COCV 12 (2006) 93–119. CrossRef