# 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

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topClason, 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

top- R.A. Adams and J.J.F. Fournier, Sobolev Spaces, Pure and Applied Mathematics (Amsterdam)140. Second edition, Elsevier/Academic Press, Amsterdam (2003).
- 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.
- T. Grund and A. Rösch, Optimal control of a linear elliptic equation with a supremum norm functional. Optim. Methods Softw.15 (2001) 299–329.
- 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.
- 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).
- 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.
- U. Prüfert and A. Schiela, The minimization of a maximum-norm functional subject to an elliptic PDE and state constraints. ZAMM89 (2009) 536–551.
- G.M. Troianiello, Elliptic Differential Equations and Obstacle Problems. The University Series in Mathematics, Plenum Press, New York (1987).

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