Skip to main content
SpringerLink
Log in

Numerical Analysis of State-constrained Optimal Control Problems for PDEs

  • Chapter
  • First Online:
Constrained Optimization and Optimal Control for Partial Differential Equations

Part of the book series: International Series of Numerical Mathematics ((ISNM,volume 160))

Abstract

We survey the results of SPP 1253 project “Numerical Analysis of State-constrained Optimal Control Problems for PDEs”. In the first part, we consider Lavrentiev-type regularization of both distributed and boundary control. In the second part, we present a priori error estimates for elliptic control problems with finite-dimensional control space and state-constraints both in finitely many points and in all points of a subdomain with nonempty interior.

Mathematics Subject Classification (2000). 49K20, 49M05, 90C06, 90C34, 90C30.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M. Bergounioux, M. Haddou, M. Hintermüller, and K. Kunisch. A comparison of a Moreau-Yosida-based active set strategy and interior point methods for constrained optimal control problems. SIAM J. Optimization, 11:495–521, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  2. J.T. Betts and S.L. Campbell. Discretize then Optimize. In D.R. Ferguson and T.J. Peters, editors, Mathematics in Industry: Challenges and Frontiers A Process View: Practice and Theory. SIAM Publications, Philadelphia, 2005.

    Google Scholar 

  3. F. Bonnans and A. Shapiro. Perturbation analysis of optimization problems. Springer, New York, 2000.

    Book  MATH  Google Scholar 

  4. E. Casas. Pontryagin’s principle for state-constrained boundary control problems of semilinear parabolic equations. SIAM J. Control and Optimization, 35:1297–1327, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  5. E. Casas. Error estimates for the numerical approximation of semilinear elliptic control problems with finitely many state constraints. ESAIM: Control, Optimization and Calculus of Variations, 31:345–374, 2002.

    MathSciNet  MATH  Google Scholar 

  6. E. Casas and M. Mateos. Uniform convergence of the FEM. Applications to state constrained control problems. J. of Computational and Applied Mathematics, 21:67–100, 2002.

    MathSciNet  MATH  Google Scholar 

  7. E. Casas, J. de los Reyes, and F. Tröltzsch. Sufficient second-order optimality conditions for semilinear control problems with pointwise state constraints. SIAM J. Optimization, 19(2):616–643, 2008.

    Article  MathSciNet  MATH  Google Scholar 

  8. J.C. de los Reyes, P. Merino, J. Rehberg, and F. Tröltzsch. Optimality conditions for state-constrained PDE control problems with time-dependent controls. Control Cybern., 37:5–38, 2008.

    MathSciNet  MATH  Google Scholar 

  9. K. Deckelnick and M. Hinze. Convergence of a finite element approximation to a state constrained elliptic control problem. SIAM J. Numer. Anal., 45:1937–1953, 2007.

    Article  MathSciNet  MATH  Google Scholar 

  10. K. Deckelnick and M. Hinze. Numerical analysis of a control and state constrained elliptic control problem with piecewise constant control approximations. In K. Kunisch, G. Of, and O. Steinbach, editors, Proceedings of ENUMATH 2007, the 7th European Conference on Numerical Mathematics and Advanced Applications, Heidelberg, September 2007 2008. Springer.

    Google Scholar 

  11. P. Deuflhard, M. Seebass, D. Stalling, R. Beck, and H.-C. Hege. Hyperthermia treatment planning in clinical cancer therapy: Modelling, simulation, and visualization. In A. Sydow, editor, Computational Physics, Chemistry and Biology, pages 9–17. Wissenschaft und Technik-Verlag, 1997.

    Google Scholar 

  12. K. Eppler and F. Tröltzsch. Fast optimization methods in the selective cooling of steel. In M. Grötschel, S.O. Krumke, and J. Rambau, editors, Online Optimization of Large Scale Systems, pages 185–204. Springer, 2001.

    Google Scholar 

  13. R. Griesse, N. Metla, and A. Rösch. Convergence analysis of the SQP method for nonlinear mixed-constrained elliptic optimal control problems. ZAMM, 88(10):776–792, 2008.

    Article  MathSciNet  MATH  Google Scholar 

  14. M. Hintermüller and M. Hinze. Moreau-Yosida regularization in state constrained elliptic control problems: Error estimates and parameter adjustment. SIAM J. on Numerical Analysis, 47:1666–1683, 2009.

    Article  MathSciNet  MATH  Google Scholar 

  15. M. Hintermüller, K. Ito, and K. Kunisch. The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim., 13:865–888, 2003.

    Article  MathSciNet  MATH  Google Scholar 

  16. M. Hintermüller and A. Schiela. Discretization of interior point methods for state constrained elliptic optimal control problems: Optimal error estimates and parameter adjustment. COAP, published online, 2009.

    Google Scholar 

  17. M. Hintermüller and K. Kunisch. Path-following methods for a class of constrained minimization problems in function space. SIAM J. on Optimization, 17:159–187, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  18. M. Hintermüller, F. Tröltzsch, and I. Yousept. Mesh-independence of semismooth Newton methods for Lavrentiev-regularized state constrained nonlinear optimal control problems. Numerische Mathematik, 108(4):571–603, 2008.

    Article  MathSciNet  MATH  Google Scholar 

  19. M. Hinze. A variational discretization concept in control constrained optimization: the linear-quadratic case. J. Computational Optimization and Applications, 30:45–63, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  20. K. Ito and K. Kunisch. Semi-smooth Newton methods for state-constrained optimal control problems. Systems and Control Letters, 50:221–228, 2003.

    Article  MathSciNet  MATH  Google Scholar 

  21. K. Krumbiegel, I. Neitzel, and A. Rösch, Regularization for semilinear elliptic optimal control problems with pointwise state and control constraints, Computational optimization and applications, online first. DOI 10.1007/s10589-010-9357-z

    Google Scholar 

  22. K. Krumbiegel and A. Rösch. A virtual control concept for state constrained optimal control problems. COAP, 43(2):213–233, 2009.

    MathSciNet  MATH  Google Scholar 

  23. P. Merino, I. Neitzel, and F. Tröltzsch. Error estimates for the finite element discretization of semi-infinite elliptic optimal control problems. Discussiones Mathematicae, Differential Inclusions, Control and Optimization, 30(2):221–236, 2010.

    Article  MathSciNet  MATH  Google Scholar 

  24. P. Merino, I. Neitzel, and F. Tröltzsch. On linear-quadratic elliptic optimal control problems of semi-infinite type. Applicable Analysis, 90(6):1047–1074, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  25. P. Merino, F. Tröltzsch, and B. Vexler. 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, 44:167–188, 2010.

    Article  MathSciNet  MATH  Google Scholar 

  26. C. Meyer. Error estimates for the finite-element approximation of an elliptic control problem with pointwise state and control constraints. Control Cybern., 37:51–85, 2008.

    MathSciNet  MATH  Google Scholar 

  27. C. Meyer and P. Philip. Optimizing the temperature profile during sublimation growth of SiC single crystals: Control of heating power, frequency, and coil position. Crystal Growth & Design, 5:1145–1156, 2005.

    Article  Google Scholar 

  28. C. Meyer, A. Rösch, and F. Tröltzsch. Optimal control of PDEs with regularized pointwise state constraints. Computational Optimization and Applications, 33(2003-14):209–228, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  29. C. Meyer and F. Tröltzsch. On an elliptic optimal control problem with pointwise mixed control-state constraints. In A. Seeger, editor, Recent Advances in Optimization. Proceedings of the 12th French-German-Spanish Conference on Optimization held in Avignon, September 20–24, 2004, Lectures Notes in Economics and Mathematical Systems. Springer-Verlag, 2005.

    Google Scholar 

  30. I. Neitzel, U. Prüfert, and T. Slawig. Strategies for time-dependent PDE control with inequality constraints using an integrated modeling and simulation environment. Numerical Algorithms, 50:241–269, 2009.

    Article  MathSciNet  MATH  Google Scholar 

  31. I. Neitzel and F. Tröltzsch. On convergence of regularization methods for nonlinear parabolic optimal control problems with control and state constraints. Control and Cybernetics, 37(4):1013–1043, 2008.

    MathSciNet  MATH  Google Scholar 

  32. I. Neitzel and F. Tröltzsch. On regularization methods for the numerical solution of parabolic control problems with pointwise state constraints. ESAIM Control, Optimisation and Calculus of Variations, 15( 2):426–453, 2009.

    Article  MathSciNet  MATH  Google Scholar 

  33. U. Prüfert, F. Tröltzsch, and M. Weiser. The convergence of an interior point method for an elliptic control problem with mixed control-state constraints. Comput. Optim. Appl., 39(2):183–218, March 2008.

    Article  MathSciNet  MATH  Google Scholar 

  34. R. Rannacher and B. Vexler. A priori error estimates for the finite element discretization of elliptic parameter identification problems with pointwise measurements. SIAM Control Optim., 44:1844–1863, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  35. J. Raymond and F. Tröltzsch. Second-order sufficient optimality conditions for nonlinear parabolic control problems with state constraints. Discrete Contin. Dyn. Syst., 6:431–450, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  36. A. Rösch and F. Tröltzsch. Existence of regular Lagrange multipliers for a nonlinear elliptic optimal control problem with pointwise control-state constraints. SIAM J. Control and Optimization, 45:548–564, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  37. A. Rösch and F. Tröltzsch. Sufficient second-order optimality conditions for an elliptic optimal control problem with pointwise control-state constraints. SIAM J. on Optimization, 17(3):776–794, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  38. A. Schiela. Barrier methods for optimal control problems with state constraints. SIAM J. on Optimization, 20:1002–1031, 2009.

    Article  MathSciNet  MATH  Google Scholar 

  39. F. Tröltzsch. Optimal Control of Partial Differential Equations: Theory, Methods and Applications. AMS, Providence, 2010.

    Google Scholar 

  40. F. Tröltzsch and I. Yousept. A regularization method for the numerical solution of elliptic boundary control problems with pointwise state constraints. COAP, 42(1):43–66, 2009.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Mathematik, Technische Universität Berlin, Str. des 17. Juni 136, D-10623, Berlin, Germany

    Ira Neitzel & Fredi Tröltzsch

Authors
  1. Ira Neitzel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fredi Tröltzsch
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ira Neitzel .

Editor information

Editors and Affiliations

  1. , Department Mathematik, Universität Erlangen-Nürnberg, Martensstr. 3, Erlangen, 91058, Germany

    Günter Leugering

  2. , Fakultät Bio- und Chemieingenieurwesen, Technische Universität Dortmund, Geschossbau 2, Dortmund, 44221, Germany

    Sebastian Engell

  3. , Institut für Mathematik, Humboldt-Universität zu Berlin, Rudower Chaussee 25, Berlin, 12489, Germany

    Andreas Griewank

  4. , Fachbereich Mathematik, Universität Hamburg, Bundesstr. 55, Hamburg, 20146, Germany

    Michael Hinze

  5. , Institut für Angewandte Mathematik, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 293, Heidelberg, 69120, Germany

    Rolf Rannacher

  6. , FB 4 - Department Mathematik, Universität Trier, Trier, 54286, Germany

    Volker Schulz

  7. Fakultät für Mathematik, M1, Lehrstuhl für Mathematische Optimierung, Technische Universität München, Boltzmannstr. 3, Garching bei München, 85748, Germany

    Michael Ulbrich

  8. , Fachbereich Mathematik, Technische Universität Darmstadt, Dolivostrasse 15, Darmstadt, 64293, Germany

    Stefan Ulbrich

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer Basel AG

About this chapter

Cite this chapter

Neitzel, I., Tröltzsch, F. (2012). Numerical Analysis of State-constrained Optimal Control Problems for PDEs. In: Leugering, G., et al. Constrained Optimization and Optimal Control for Partial Differential Equations. International Series of Numerical Mathematics, vol 160. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0133-1_24

Download citation

Publish with us

Policies and ethics

Search

Navigation