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Minimal invasion: An optimal L∞ state constraint problem

Published online by Cambridge University Press:  11 October 2010

Christian Clason ,
Kazufumi Ito  and
Karl Kunisch
Christian Clason
Affiliation:
Institute for Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, 8010 Graz, Austria. christian.clason@uni-graz.at; karl.kunisch@uni-graz.at
Kazufumi Ito
Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, North Carolina, 27695-8205, USA. kito@math.ncsu.edu
Karl Kunisch
Affiliation:
Institute for Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, 8010 Graz, Austria. christian.clason@uni-graz.at; karl.kunisch@uni-graz.at
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Abstract

In this work, the least pointwise upper and/or lower bounds on the state variableon 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-Yosidaregularization of the state constraints, the problem can be solvedusing a superlinearly convergent semi-smooth Newton method.Optimality conditions are derived, convergence of the Moreau-Yosidaregularization is proved, and well-posedness and superlinearconvergence of the Newton method is shown. Numerical examplesillustrate the features of this problem and the proposed approach.

Keywords

Optimal controloptimal L∞ state constraintsemi-smooth Newton method
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 45 , Issue 3 , May 2011 , pp. 505 - 522
Copyright
© EDP Sciences, SMAI, 2010

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References

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.
Grund, T. and Rösch, A., Optimal control of a linear elliptic equation with a supremum norm functional. Optim. Methods Softw. 15 (2001) 299–329. CrossRef
Hintermüller, M. and Kunisch, K., Path-following methods for a class of constrained minimization problems in function space. SIAM J. Optim. 17 (2006) 159–187. CrossRef
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).
Maurer, H. and Zowe, J., First and second order necessary and sufficient optimality conditions for infinite-dimensional programming problems. Math. Program. 16 (1979) 98–110. CrossRef
Prüfert, U. and Schiela, A., The minimization of a maximum-norm functional subject to an elliptic PDE and state constraints. ZAMM 89 (2009) 536–551. CrossRef
G.M. Troianiello, Elliptic Differential Equations and Obstacle Problems. The University Series in Mathematics, Plenum Press, New York (1987).