Minimal invasion: An optimal L∞ state constraint problem

Christian Clason; Kazufumi Ito; Karl Kunisch

ESAIM: Mathematical Modelling and Numerical Analysis (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 45.3 (2011): 505-522. <http://eudml.org/doc/197401>.

@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},
keywords = {Optimal control; optimal L∞ state constraint; semi-smooth Newton method; optimal control; optimal state constraint; Moreau-Yosida regularization; superlinear convergence; numerical examples},
language = {eng},
month = {1},
number = {3},
pages = {505-522},
publisher = {EDP Sciences},
title = {Minimal invasion: An optimal L∞ state constraint problem},
url = {http://eudml.org/doc/197401},
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
DA - 2011/1//
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 control; optimal state constraint; Moreau-Yosida regularization; superlinear convergence; numerical examples
UR - http://eudml.org/doc/197401
ER -

References

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  1. R.A. Adams and J.J.F. Fournier, Sobolev Spaces, Pure and Applied Mathematics (Amsterdam)140. Second edition, Elsevier/Academic Press, Amsterdam (2003).  
  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.  Zbl1042.35002
  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.  Zbl1005.49013
  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.  Zbl1137.49028
  5. K. Ito and K. Kunisch, Lagrange Multiplier Approach to Variational Problems and Applications, Advances in Design and Control15. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2008).  Zbl1156.49002
  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.  Zbl0398.90109
  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.  Zbl1166.49021
  8. G.M. Troianiello, Elliptic Differential Equations and Obstacle Problems. The University Series in Mathematics, Plenum Press, New York (1987).  Zbl0655.35002

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