A minimum effort optimal control problem for elliptic PDEs

Christian Clason; Kazufumi Ito; Karl Kunisch

ESAIM: Mathematical Modelling and Numerical Analysis (2012)

  • Volume: 46, Issue: 4, page 911-927
  • ISSN: 0764-583X

Abstract

top
This work is concerned with a class of minimum effort problems for partial differential equations, where the control cost is of L∞-type. Since this problem is non-differentiable, a regularized functional is introduced that can be minimized by a superlinearly convergent semi-smooth Newton method. Uniqueness and convergence for the solutions to the regularized problem are addressed, and a continuation strategy based on a model function is proposed. Numerical examples for a convection-diffusion equation illustrate the behavior of minimum effort controls.

How to cite

top

Clason, Christian, Ito, Kazufumi, and Kunisch, Karl. "A minimum effort optimal control problem for elliptic PDEs." ESAIM: Mathematical Modelling and Numerical Analysis 46.4 (2012): 911-927. <http://eudml.org/doc/222120>.

@article{Clason2012,
abstract = {This work is concerned with a class of minimum effort problems for partial differential equations, where the control cost is of L∞-type. Since this problem is non-differentiable, a regularized functional is introduced that can be minimized by a superlinearly convergent semi-smooth Newton method. Uniqueness and convergence for the solutions to the regularized problem are addressed, and a continuation strategy based on a model function is proposed. Numerical examples for a convection-diffusion equation illustrate the behavior of minimum effort controls.},
author = {Clason, Christian, Ito, Kazufumi, Kunisch, Karl},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Optimal control; minimum effort; L∞control cost; semi-smooth Newton method; optimal control; control cost},
language = {eng},
month = {2},
number = {4},
pages = {911-927},
publisher = {EDP Sciences},
title = {A minimum effort optimal control problem for elliptic PDEs},
url = {http://eudml.org/doc/222120},
volume = {46},
year = {2012},
}

TY - JOUR
AU - Clason, Christian
AU - Ito, Kazufumi
AU - Kunisch, Karl
TI - A minimum effort optimal control problem for elliptic PDEs
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2012/2//
PB - EDP Sciences
VL - 46
IS - 4
SP - 911
EP - 927
AB - This work is concerned with a class of minimum effort problems for partial differential equations, where the control cost is of L∞-type. Since this problem is non-differentiable, a regularized functional is introduced that can be minimized by a superlinearly convergent semi-smooth Newton method. Uniqueness and convergence for the solutions to the regularized problem are addressed, and a continuation strategy based on a model function is proposed. Numerical examples for a convection-diffusion equation illustrate the behavior of minimum effort controls.
LA - eng
KW - Optimal control; minimum effort; L∞control cost; semi-smooth Newton method; optimal control; control cost
UR - http://eudml.org/doc/222120
ER -

References

top
  1. J.Z. Ben-Asher, E.M. Cliff and J.A. Burns, Computational methods for the minimum effort problem with applications to spacecraft rotational maneuvers, in IEEE Conf. on Control and Applications (1989) 472–478.  
  2. C. Clason, K. Ito and K. Kunisch, Minimal invasion : An optimal L∞ state constraint problem. ESAIM : M2AN45 (2010) 505–522.  Zbl1269.65060
  3. I. Ekeland and R. Témam, Convex analysis and variational problems, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA (1999).  Zbl0939.49002
  4. 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
  5. 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
  6. M. Gugat and G. Leugering, L∞-norm minimal control of the wave equation : On the weakness of the bang-bang principle. ESAIM Control Optim. Calc. Var.14 (2008) 254–283.  Zbl1133.49006
  7. K. Ito and K. Kunisch, Lagrange Multiplier Approach to Variational Problems and Applications, Advances in Design and Control. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA 15 (2008).  Zbl1156.49002
  8. K. Ito and K. Kunisch, Minimal effort problems and their treatment by semismooth newton methods. SIAM J. Control Optim.49 (2011) 2083–2100.  Zbl1234.49017
  9. O.A. Ladyzhenskaya and N.N. Ural’tseva, Linear and quasilinear elliptic equations. Translated from the Russian by Scripta Technica, Inc. Translation, edited by L. Ehrenpreis, Academic Press, New York (1968).  
  10. L.W. Neustadt, Minimum effort control systems. SIAM J. Control Ser. A1 (1962) 16–31.  Zbl0145.34502
  11. U. Prüfert and A. Schiela, The minimization of a maximum-norm functional subject to an elliptic PDE and state constraints. Z. Angew. Math. Mech.89 (2009) 536–551.  Zbl1166.49021
  12. A. Schiela, A simplified approach to semismooth Newton methods in function space. SIAM J. Optim.19 (2008) 1417–1432.  Zbl1169.49032
  13. Z. Sun and J. Zeng, A damped semismooth Newton method for mixed linear complementarity problems. Optim. Methods Softw.26 (2010) 187–205.  Zbl1251.90369
  14. G.M. Troianiello, Elliptic Differential Equations and Obstacle Problems. The University Series in Mathematics, Plenum Press, New York (1987).  Zbl0655.35002
  15. F. Tröltzsch, Optimal Control of Partial Differential Equations : Theory, Methods and Applications. American Mathematical Society, Providence (2010). Translated from the German by Jürgen Sprekels.  Zbl1195.49001
  16. E. Zuazua, Controllability and observability of partial differential equations : some results and open problems, in Handbook of differential equations : evolutionary equations. Handb. Differ. Equ., Elsevier, North, Holland, Amsterdam III (2007) 527–621.  Zbl1193.35234

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.