Minimal invasion: An optimal L ∞ state constraint problem
Clason, Christian ; Ito, Kazufumi ; Kunisch, Karl
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 45 (2011), p. 505-522 / Harvested from Numdam
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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.

Publié le : 2011-01-01
DOI : https://doi.org/10.1051/m2an/2010064
Classification:  49J52,  49J20,  49K20 
@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 - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     volume = {45},
     year = {2011},
     pages = {505-522},
     doi = {10.1051/m2an/2010064},
     zbl = {1269.65060},
     language = {en},
     url = {http://dml.mathdoc.fr/item/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 - Modélisation Mathématique et Analyse Numérique, Tome 45 (2011) pp. 505-522. doi : 10.1051/m2an/2010064. http://gdmltest.u-ga.fr/item/M2AN_2011__45_3_505_0/

[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 2424078 | Zbl 1098.46001

[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 1814364 | Zbl 1042.35002

[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 1892589 | Zbl 1005.49013

[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 2219149 | Zbl 1137.49028

[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 2441683 | Zbl 1156.49002

[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 517762 | Zbl 0398.90109

[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 2553754 | Zbl 1166.49021

[8] G.M. Troianiello, Elliptic Differential Equations and Obstacle Problems. The University Series in Mathematics, Plenum Press, New York (1987). | MR 1094820 | Zbl 0655.35002