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

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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.

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References

  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)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bergounioux, M., Ito, K., Kunisch, K.: Primal-dual strategy for constrained optimal control problems. SIAM J. Control Optim. 37, 1176–1194 (1999)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  6. Gruver, W.A., Sachs, E.W.: Algorithmic Methods in Optimal Control. Pitman, London (1980)

    Google Scholar 

  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)

    Article  MATH  Google Scholar 

  8. Ito, K., Kunisch, K.: Semi-smooth Newton methods for state-constrained optimal control problems. Syst. Control. Lett. 50, 221–228 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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)

    Article  MATH  MathSciNet  Google Scholar 

  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. Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)

    MATH  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  14. Meyer, C., Rösch, A., Tröltzsch, F.: Optimal control of PDEs with regularized pointwise state constraints. Comput. Optim. Appl. 33, 209–228 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  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. MOSEK ApS: The MOSEK optimization tools manual. Version 5.0 (Revision 60). http://www.mosek.com (2007)

  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)

  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)

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  21. Schiela, A.: Barrier Methods for Optimal Control Problems with State Constraints. ZIB-Report 07-07, Konrad-Zuse-Zentrum für Informationstechnik, Berlin (2007)

  22. The MathWorks: Partial Differential Equation Toolbox User’s Guide. The Math Works Inc., Natick (1995)

  23. Tröltzsch, F.: Optimale Steuerung partieller Differentialgleichungen. Theorie, Verfahren und Anwendungen. Vieweg, Wiesbaden (2005)

    MATH  Google Scholar 

  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. Ulbrich, M., Ulbrich, S.: Primal-dual interior point methods for PDE-constrained optimization. Math. Program. doi:10.1007/s10107-007-0168-7 (2007)

  26. Wloka, J.: Partielle Differentialgleichungen. Teubner-Verlag, Leipzig (1982)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Fakultät II Institut für Mathematik, TU Berlin, Straße des 17. Juni 136, 10623, Berlin, Germany

    Ira Neitzel & Uwe Prüfert

  2. Technische Fakultät, Christian-Albrechts-Universität zu Kiel, Christian-Albrechts-Platz 4, 24118, Kiel, Germany

    Thomas Slawig

Authors
  1. Ira Neitzel
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  2. Uwe Prüfert
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  3. Thomas Slawig
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Corresponding author

Correspondence to Uwe Prüfert.

Additional information

Ira Neitzel’s research was supported by the DFG Schwerpunktprogramm SPP 1253.

Uwe Prüfert’s research was supported by the DFG Research Center Matheon.

Thomas Slawig’s research was supported by the DFG Cluster of Excellence The Future Ocean and the DFG Schwerpunktprogramm SPP 1253.

Website www.math.tu-berlin.de/Strategies-for-time-dependent-PDE-control.html

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Neitzel, I., Prüfert, U. & Slawig, T. Strategies for time-dependent PDE control with inequality constraints using an integrated modeling and simulation environment. Numer Algor 50, 241–269 (2009). https://doi.org/10.1007/s11075-008-9225-4

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  • DOI: https://doi.org/10.1007/s11075-008-9225-4

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