Error estimates for the finite element approximation of a semilinear elliptic control problem with state constraints and finite dimensional control space

Pedro Merino; Fredi Tröltzsch; Boris Vexler

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 44, Issue: 1, page 167-188
  • ISSN: 0764-583X

Abstract

top
The finite element approximation of optimal control problems for semilinear elliptic partial differential equation is considered, where the control belongs to a finite-dimensional set and state constraints are given in finitely many points of the domain. Under the standard linear independency condition on the active gradients and a strong second-order sufficient optimality condition, optimal error estimates are derived for locally optimal controls.

How to cite

top

Merino, Pedro, Tröltzsch, Fredi, and Vexler, Boris. "Error estimates for the finite element approximation of a semilinear elliptic control problem with state constraints and finite dimensional control space." ESAIM: Mathematical Modelling and Numerical Analysis 44.1 (2010): 167-188. <http://eudml.org/doc/250820>.

@article{Merino2010,
abstract = { The finite element approximation of optimal control problems for semilinear elliptic partial differential equation is considered, where the control belongs to a finite-dimensional set and state constraints are given in finitely many points of the domain. Under the standard linear independency condition on the active gradients and a strong second-order sufficient optimality condition, optimal error estimates are derived for locally optimal controls. },
author = {Merino, Pedro, Tröltzsch, Fredi, Vexler, Boris},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Finite element approximation; optimal control problem; finitely many pointwise state constraints; finite element; optimal control; semilinear elliptic partial differential equation; optimal error estimates},
language = {eng},
month = {3},
number = {1},
pages = {167-188},
publisher = {EDP Sciences},
title = {Error estimates for the finite element approximation of a semilinear elliptic control problem with state constraints and finite dimensional control space},
url = {http://eudml.org/doc/250820},
volume = {44},
year = {2010},
}

TY - JOUR
AU - Merino, Pedro
AU - Tröltzsch, Fredi
AU - Vexler, Boris
TI - Error estimates for the finite element approximation of a semilinear elliptic control problem with state constraints and finite dimensional control space
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 44
IS - 1
SP - 167
EP - 188
AB - The finite element approximation of optimal control problems for semilinear elliptic partial differential equation is considered, where the control belongs to a finite-dimensional set and state constraints are given in finitely many points of the domain. Under the standard linear independency condition on the active gradients and a strong second-order sufficient optimality condition, optimal error estimates are derived for locally optimal controls.
LA - eng
KW - Finite element approximation; optimal control problem; finitely many pointwise state constraints; finite element; optimal control; semilinear elliptic partial differential equation; optimal error estimates
UR - http://eudml.org/doc/250820
ER -

References

top
  1. E.L. Allgower, K. Böhmer, F.A. Potra and W.C. Rheinboldt, A mesh-independence principle for operator equations and their discretizations. SIAM J. Numer. Anal.23 (1986) 160–169.  
  2. W. Alt, On the approximation of infinite optimization problems with an application to optimal control problems. Appl. Math. Opt.12 (1984) 15–27.  
  3. N. Arada, E. Casas and F. Tröltzsch, Error estimates for the numerical approximation of a semilinear elliptic control problem. Comput. Optim. Appl.23 (2002) 201–229.  
  4. A. Bermúdez, P. Gamallo and R. Rodríguez, Finite element methods in local active control of sound. SIAM J. Control Optim.43 (2004) 437–465 (electronic).  
  5. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems. Springer-Verlag, New York, USA (2000).  
  6. S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer, New York, USA (1994).  
  7. E. Casas, Boundary control of semilinear elliptic equations with pointwise state constraints. SIAM J. Control Optim.31 (1993) 993–1006.  
  8. E. Casas, Error estimates for the numerical approximation of semilinear elliptic control problems with finitely many state contraints. ESAIM: COCV8 (2002) 345–374.  
  9. E. Casas, Using piecewise linear functions in the numerical approximation of semilinear elliptic control problems. Adv. Comput. Math.26 (2007) 137–153.  
  10. E. Casas and M. Mateos, Second order sufficient optimality conditions for semilinear elliptic control problems with finitely many state constraints. SIAM J. Control Optim.40 (2002) 1431–1454.  
  11. E. Casas and M. Mateos, Uniform convergence of the FEM. Applications to state constrained control problems. J. Comput. Appl. Math.21 (2002) 67–100.  
  12. J.C. de los Reyes, P. Merino, J. Rehberg and F. Tröltzsch, Optimality conditions for state-constrained PDE control problems with finite-dimensional control space. Control Cybern. (to appear).  
  13. M. Deckelnick and M. Hinze, Convergence of a finite element approximation to a state constrained elliptic control problem. SIAM J. Numer. Anal.45 (2007) 1937–1953.  
  14. M. Deckelnick and M. Hinze, Numerical analysis of a control and state constrained elliptic control problem with piecewise constant control approximations, in Numerical Mathematics and Advanced Applications, Proc. of ENUMATH 2007, Graz, K. Kunisch, G. Of and O. Steinbach Eds., Springer, Berlin-Heidelberg, Germany (2008) 597–604.  
  15. A.V. Fiacco and G.P. McCormick, Nonlinear Programming: Sequential Unconstrained Minimization Techniques. J. Wiley and Sons, Inc., New York, USA (1968).  
  16. J. Frehse and R. Rannacher, Eine l1-Fehlerabschätzung diskreter Grundlösungen in der Methode der finiten Elemente. Bonner Math. Schriften89 (1976) 92–114.  
  17. D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer, Berlin, Germany (1998).  
  18. P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman, Boston, USA (1985).  
  19. D. Klatte, A note on quantitative stability results in nonlinear optimization. Seminarbericht 90, Humboldt-Universität zu Berlin, Sektion Mathematik, Germany (1987).  
  20. D. Klatte and B. Kummer, Nonsmooth Equations in Optimization: Regularity, Calculus, Methods and Applications. Kluwer Academic Publishers, Dordrecht, The Netherlands (2002).  
  21. D.G. Luenberger, Linear and Nonlinear Programming. Addison Wesley, Reading, Massachusetts, USA (1984).  
  22. K. Malanowski, Stability of solutions to convex problems of optimization, Lecture Notes Contr. Inf. Sci.93, Springer-Verlag, Berlin, Germany (1987).  
  23. K. Malanowski, Ch. Büskens and H. Maurer, Convergence of approximations to nonlinear optimal control problems, in Mathematical Programming with Data Perturbations, A.V. Fiacco Ed., Lecture Notes to Pure and Applied Mathematics195, Marcel Dekker, New York, USA (1998) 253–284.  
  24. C. Meyer, Error estimates for the finite-element approximation of an elliptic control problem with pointwise state and control constraints. Contr. Cybern.37 (2008) 51–85.  
  25. C. Meyer, U. Prüfert and F. Tröltzsch, On two numerical methods for state-constrained elliptic control problems. Otim. Meth. Software22 (2007) 871–899.  
  26. R. Rannacher, Zur l -Konvergenz linearer finiter Elemente beim Dirichlet-Problem. Math. Z.149 (1976) 69–77.  
  27. R. Rannacher and B. Vexler, A priori error estimates for the finite element discretization of elliptic parameter identification problems with pointwise measurements. SIAM J. Control Optim.44 (2005) 1844–1863.  
  28. S.M. Robinson, Stability theory for systems of inequalities, II: Differentiable nonlinear systems. SIAM J. Numer. Anal.13 (1976) 497–513.  
  29. S.M. Robinson, Strongly regular generalized equations. Math. Oper. Res.5 (1980) 43–62.  
  30. A. Rösch, Error estimates for linear-quadratic control problems with control constraints. Optim. Methods Softw.21 (2006) 121–134.  
  31. A.H. Schatz and L.B. Wahlbin, Interior maximum norm estimates for finite element methods. Math. Comp.31 (1977) 414–442.  
  32. A.H. Schatz and L.B. Wahlbin, Interior maximum-norm estimates for finite element methods, part II. Math. Comp.64 (1995) 907–928.  
  33. F. Tröltzsch, Optimale Steuerung partieller Differentialgleichungen – Theorie, Verfahren und Anwendungen. Vieweg, Wiesbaden, Germany (2005).  
  34. F. Tröltzsch, On finite element error estimates for optimal control problems with elliptic PDEs, in The Proceedings of the Conference on Large Scale Scientific Computing, Sozopol, Bulgaria, June 4–8, 2009, Lect. Notes in Comp. Sci., Springer-Verlag (to appear).  
  35. J. Zowe and S. Kurcyusz, Regularity and stability for the mathematical programming problem in Banach spaces. Appl. Math. Optim.5 (1979) 49–62.  

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.