no. 3
Minimal invasion: An optimal L ∞ state constraint problem
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 45 (2011) no. 3, pp. 505-522.

In this work, the least pointwise upper and/or lower bounds on the state variable on a specified subdomain of a control system under piecewise constant control action are sought. This results in a non-smooth optimization problem in function spaces. Introducing a Moreau-Yosida regularization of the state constraints, the problem can be solved using a superlinearly convergent semi-smooth Newton method. Optimality conditions are derived, convergence of the Moreau-Yosida regularization is proved, and well-posedness and superlinear convergence of the Newton method is shown. Numerical examples illustrate the features of this problem and the proposed approach.

DOI : 10.1051/m2an/2010064
Classification : 49J52, 49J20, 49K20
Mots clés : optimal control, optimal L∞ state constraint, semi-smooth Newton method 
@article{M2AN_2011__45_3_505_0,
     author = {Clason, Christian and Ito, Kazufumi and Kunisch, Karl},
     title = {Minimal invasion: {An} optimal $L^\infty $ state constraint problem},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {505--522},
     publisher = {EDP-Sciences},
     volume = {45},
     number = {3},
     year = {2011},
     doi = {10.1051/m2an/2010064},
     zbl = {1269.65060},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2010064/}
}
TY  - JOUR
AU  - Clason, Christian
AU  - Ito, Kazufumi
AU  - Kunisch, Karl
TI  - Minimal invasion: An optimal $L^\infty $ state constraint problem
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2011
SP  - 505
EP  - 522
VL  - 45
IS  - 3
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2010064/
DO  - 10.1051/m2an/2010064
LA  - en
ID  - M2AN_2011__45_3_505_0
ER  - 
%0 Journal Article
%A Clason, Christian
%A Ito, Kazufumi
%A Kunisch, Karl
%T Minimal invasion: An optimal $L^\infty $ state constraint problem
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2011
%P 505-522
%V 45
%N 3
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2010064/
%R 10.1051/m2an/2010064
%G en
%F M2AN_2011__45_3_505_0
Clason, Christian; Ito, Kazufumi; Kunisch, Karl. Minimal invasion: An optimal $L^\infty $ state constraint problem. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 45 (2011) no. 3, pp. 505-522. doi : 10.1051/m2an/2010064. http://www.numdam.org/articles/10.1051/m2an/2010064/

[1] R.A. Adams and J.J.F. Fournier, Sobolev Spaces, Pure and Applied Mathematics (Amsterdam) 140. Second edition, Elsevier/Academic Press, Amsterdam (2003). | MR | Zbl

[2] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Classics in Mathematics, Springer-Verlag, Berlin (2001). Reprint of the 1998 edition. | MR | Zbl

[3] T. Grund and A. Rösch, Optimal control of a linear elliptic equation with a supremum norm functional. Optim. Methods Softw. 15 (2001) 299-329. | MR | Zbl

[4] M. Hintermüller and K. Kunisch, Path-following methods for a class of constrained minimization problems in function space. SIAM J. Optim. 17 (2006) 159-187. | MR | Zbl

[5] K. Ito and K. Kunisch, Lagrange Multiplier Approach to Variational Problems and Applications, Advances in Design and Control 15. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2008). | MR | Zbl

[6] H. Maurer and J. Zowe, First and second order necessary and sufficient optimality conditions for infinite-dimensional programming problems. Math. Program. 16 (1979) 98-110. | MR | Zbl

[7] U. Prüfert and A. Schiela, The minimization of a maximum-norm functional subject to an elliptic PDE and state constraints. ZAMM 89 (2009) 536-551. | MR | Zbl

[8] G.M. Troianiello, Elliptic Differential Equations and Obstacle Problems. The University Series in Mathematics, Plenum Press, New York (1987). | MR | Zbl

Cité par Sources :