Numerical Algorithms

, Volume 50, Issue 3, pp 241–269 | Cite as

Strategies for time-dependent PDE control with inequality constraints using an integrated modeling and simulation environment

Original Paper

Abstract

In Neitzel et al. (Strategies for time-dependent PDE control using an integrated modeling and simulation environment. Part one: problems without inequality constraints. Technical Report 408, Matheon, Berlin, 2007) we have shown how time-dependent optimal control for partial differential equations can be realized in a modern high-level modeling and simulation package. In this article we extend our approach to (state) constrained problems. “Pure” state constraints in a function space setting lead to non-regular Lagrange multipliers (if they exist), i.e. the Lagrange multipliers are in general Borel measures. This will be overcome by different regularization techniques. To implement inequality constraints, active set methods and barrier methods are widely in use. We show how these techniques can be realized in a modeling and simulation package. We implement a projection method based on active sets as well as a barrier method and a Moreau Yosida regularization, and compare these methods by a program that optimizes the discrete version of the given problem.

Keywords

Optimal control Parabolic PDEs Inequality constraints Integrated modeling and simulation environments 

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References

  1. 1.
    Bergounioux, M., Haddou, M., Hintermüller, M., Kunisch, K.: A comparison of a Moreau-Yosida-based active set strategy and interior point methods for constrained optimal control problems. SIAM J. Optim. 11, 495–521 (2000)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bergounioux, M., Ito, K., Kunisch, K.: Primal-dual strategy for constrained optimal control problems. SIAM J. Control Optim. 37, 1176–1194 (1999)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Betts, J.T., Campbell, S.L.: Discretize then optimize. In: Ferguson, D.R., Peters, T.J. (eds.) Mathematics in Industry: Challenges and Frontiers A Process View: Practice and Theory. SIAM, Philadelphia (2005)Google Scholar
  4. 4.
    Casas, E.: Pontryagin’s principle for state-constrained boundary control problems of semilinear parabolic equations. SIAM J. Control Optim. 35, 1297–1327 (1997)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Deckelnick, K., Hinze, M.: Convergence of a finite element approximation to a state constrained elliptic control problem. SIAM J. Numer. Anal. 45, 1937–1953 (2007)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Gruver, W.A., Sachs, E.W.: Algorithmic Methods in Optimal Control. Pitman, London (1980)Google Scholar
  7. 7.
    Hintermüller, M., Ito, K., Kunisch, K.: The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim. 13, 865–888 (2003)MATHCrossRefGoogle Scholar
  8. 8.
    Ito, K., Kunisch, K.: Semi-smooth Newton methods for state-constrained optimal control problems. Syst. Control. Lett. 50, 221–228 (2003)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Krumbiegel, K., Rösch, A.: A virtual control concept for state constrained optimal control problems. Comput. Optim. Appl. doi:10.1007/s10589-007-9130-0 (2007)
  10. 10.
    Kunisch, K., Rösch, A.: Primal-dual active set strategy for a general class of constrained optimal control problems. SIAM J. Optim. 13, 321–334 (2002)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Ladyzhenskaya, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and Quasilinear Equations of Parabolic Type. American Mathematical Society, Providence (1968)Google Scholar
  12. 12.
    Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)MATHGoogle Scholar
  13. 13.
    Meyer, C., Prüfert, U., Tröltzsch, F.: On two numerical methods for state-constrained elliptic control problems. Optim. Methods Softw. 22(6), 871–899 (2007)MATHMathSciNetGoogle Scholar
  14. 14.
    Meyer, C., Rösch, A., Tröltzsch, F.: Optimal control of PDEs with regularized pointwise state constraints. Comput. Optim. Appl. 33, 209–228 (2006)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Meyer, C., Tröltzsch, F.: On an elliptic optimal control problem with pointwise mixed control-state constraints. In: Seeger, A. (ed.) Recent Advances in Optimization. Proceedings of the 12th French-German-Spanish Conference on Optimization held in Avignon, September 20–24, 2004. Springer, Berlin (2005)Google Scholar
  16. 16.
    MOSEK ApS: The MOSEK optimization tools manual. Version 5.0 (Revision 60). http://www.mosek.com (2007)
  17. 17.
    Neitzel, I., Prüfert, U., Slawig, T.: Strategies for time-dependent PDE control using an integrated modeling and simulation environment. Part one: problems without inequality constraints. Technical Report 408, Matheon, Berlin (2007)Google Scholar
  18. 18.
    Neitzel, I., Tröltzsch, F.: On regularization methods for the numerical solution of parabolic control problems with pointwise state constraints. Technical report, SPP 1253. To appear in ESAIM: COCV (2007)Google Scholar
  19. 19.
    Prüfert, U., Tröltzsch, F.: An interior point method for a parabolic optimal control problem with regularized pointwise state constraints. ZAMM 87(8–9), 564–589 (2007)MATHCrossRefGoogle Scholar
  20. 20.
    Raymond, J.-P., Zidani, H.: Pontryagin’s principle for state-constrained control problems governed by parabolic equations with unbounded controls. SIAM J. Control Optim. 36, 1853–1879 (1998)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Schiela, A.: Barrier Methods for Optimal Control Problems with State Constraints. ZIB-Report 07-07, Konrad-Zuse-Zentrum für Informationstechnik, Berlin (2007)Google Scholar
  22. 22.
    The MathWorks: Partial Differential Equation Toolbox User’s Guide. The Math Works Inc., Natick (1995)Google Scholar
  23. 23.
    Tröltzsch, F.: Optimale Steuerung partieller Differentialgleichungen. Theorie, Verfahren und Anwendungen. Vieweg, Wiesbaden (2005)MATHGoogle Scholar
  24. 24.
    Tröltzsch, F., Yousept, I.: A regularization method for the numerical solution of elliptic boundary control problems with pointwise state constraints. Comput. Optim. Appl. doi:10.1007/s10589-007-9114-0 (2008)
  25. 25.
    Ulbrich, M., Ulbrich, S.: Primal-dual interior point methods for PDE-constrained optimization. Math. Program. doi:10.1007/s10107-007-0168-7 (2007)
  26. 26.
    Wloka, J.: Partielle Differentialgleichungen. Teubner-Verlag, Leipzig (1982)MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2008

Authors and Affiliations

  1. 1.Fakultät II Institut für MathematikTU BerlinBerlinGermany
  2. 2.Technische FakultätChristian-Albrechts-Universität zu KielKielGermany

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