On stability analysis in shape optimisation : critical shapes for Neumann problem

• Autorzy: Dambrine M. , Sokolowski J. , Żochowski A.
• Języki publikacji: EN

Abstrakty

EN

The stability issue of critical shapes for shape optimization problems with the state function given by a solution to the Neumann problem for the Laplace equation is considered. To this end, the properties of the shape Hessian evaluated at critical shapes are analysed. First, it is proved that the stability cannot be expected for the model problem. Then, the new estimates for the shape Hessian are derived in order to overcome the classical two norms-discrepancy well know in control problems, Malanowski (2001). In the context of shape optimization, the situation is similar compared to control problems, actually, the shape Hessian can be coercive only in the norm strictly weaker with respect to the norm of the second order differentiability of the shape functional. In addition, it is shown that an appropriate regularization makes possible the stability of critical shapes.

Słowa kluczowe

EN

shape optimisation; stability analysis; shape gradient; shape Hessian

PL

optymalizacja kształtu; analiza stabilności; gradient kształtu

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2003
• Tom: Vol. 32, no. 3
• Strony: 503--528
• Opis fizyczny: Bibliogr. 12 poz.

Twórcy

autor

Dambrine M.

• Laboratoire de Mathematiques Appliqu´ees de Compiegne, Universite de Technologie de Compiegne, former affiliation : Antenne de Bretagne de l’Ecole Normale Sup´erieure de Cachan where this work was done;

autor

Sokolowski J.

• Institut Elie Cartan, Laboratoire de Mathematiques, Universite Henri Poincare Nancy I, B.P. 239, 54506 Vandoeuvre les Nancy Cedex, France

autor

Żochowski A.

• Systems Research Institute of the Polish Academy of Sciences, ul. Newelska 6, 01-447 Warszawa, Poland

Bibliografia

• Belov, S. and Fuji, N. (1997) Symmetry and sufficient condition of optimality in a domain optimization problem. Control and Cybernetics, 26(1) 45–56.
• Courant, R.and Hilbert, D. (1989) Methods of mathematical physics. Vol. II. Partial differential equations. Reprint of the 1962 original. A Wiley Interscience Publication, John Wiley & Sons Inc., New York.
• Dambrine, M.(2002) About the variations of the shape Hessian and sufficient conditions of stability for critical shapes. Revista Real Academia Ciencias, RACSAM, 96(1).
• Dambrine, M. (2000) Hessiennes de forme et stabilit´e des formes critiques. PhD thesis, Ecole Normale Sup´erieure de Cachan, ´
• Dambrine, M. and Pierre, M. (2000) About stability of equilibrium shapes. Mathematical Modelling and Numerical Analysis, 34(4), 811–834.
• Descloux, J. (1990) The bidimensionnal shaping problem without surface tension. Technical report, Ecole Polytechnique F´ed´erale de Lausanne. ´
• Eppler., K. (2000) Second derivatives and suffient optimality conditions for shape functionals. Control and Cybernetics, 29(2), 485–511.
• Gilbarg, D. and Trudinger, N.S. (1983) Elliptic partial differential equations of second order, Springer-Verlag, Berlin, second edition.
• Henrot, A. and Pierre, M. (1989) Un probl`eme inverse en formage des m´etaux liquides. RAIRO Mod´el. Math. Anal. Num´er., 23(1), 155–177.
• Malanowski, K. (2001) Stability and sensitivity analysis for optimal control problems with control-state constraints. Dissertationes Math. (Rozprawy Mat.) 394.
• Murat, F. and Simon J. (1977) Optimal control with respect to the domain. These de l’Universit´e Paris VI.
• Sokolowski, J. and Zolesio, J.-P. (1992) Shape sensitivity analysis. Introduction to shape optimization. Springer-Verlag, Berlin