Minimal invasion: An optimal L∞ state constraint problem

Christian Clason; Kazufumi Ito; Karl Kunisch

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2011)

  • Volume: 45, Issue: 3, page 505-522
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Clason, Christian, Ito, Kazufumi, and Kunisch, Karl. "Minimal invasion: An optimal L∞ state constraint problem." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 45.3 (2011): 505-522. <http://eudml.org/doc/273284>.

@article{Clason2011,
abstract = {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.},
author = {Clason, Christian, Ito, Kazufumi, Kunisch, Karl},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {optimal control; optimal L∞ state constraint; semi-smooth Newton method; optimal state constraint; Moreau-Yosida regularization; superlinear convergence; numerical examples},
language = {eng},
number = {3},
pages = {505-522},
publisher = {EDP-Sciences},
title = {Minimal invasion: An optimal L∞ state constraint problem},
url = {http://eudml.org/doc/273284},
volume = {45},
year = {2011},
}

TY - JOUR
AU - Clason, Christian
AU - Ito, Kazufumi
AU - Kunisch, Karl
TI - Minimal invasion: An optimal L∞ state constraint problem
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2011
PB - EDP-Sciences
VL - 45
IS - 3
SP - 505
EP - 522
AB - 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.
LA - eng
KW - optimal control; optimal L∞ state constraint; semi-smooth Newton method; optimal state constraint; Moreau-Yosida regularization; superlinear convergence; numerical examples
UR - http://eudml.org/doc/273284
ER -

References

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  1. [1] R.A. Adams and J.J.F. Fournier, Sobolev Spaces, Pure and Applied Mathematics (Amsterdam) 140. Second edition, Elsevier/Academic Press, Amsterdam (2003). Zbl1098.46001MR2424078
  2. [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. Zbl1042.35002MR1814364
  3. [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. Zbl1005.49013MR1892589
  4. [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. Zbl1137.49028MR2219149
  5. [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). Zbl1156.49002MR2441683
  6. [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. Zbl0398.90109MR517762
  7. [7] U. Prüfert and A. Schiela, The minimization of a maximum-norm functional subject to an elliptic PDE and state constraints. ZAMM89 (2009) 536–551. Zbl1166.49021MR2553754
  8. [8] G.M. Troianiello, Elliptic Differential Equations and Obstacle Problems. The University Series in Mathematics, Plenum Press, New York (1987). Zbl0655.35002MR1094820

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