Skip to main content
SpringerLink
Log in

Error estimates for parabolic optimal control problems with control and state constraints

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

The numerical approximation to a parabolic control problem with control and state constraints is studied in this paper. We use standard piecewise linear and continuous finite elements for the space discretization of the state, while the dG(0) method is used for time discretization. A priori error estimates for control and state are obtained by an improved maximum error estimate for the corresponding discretized state equation. Numerical experiments are provided which support our theoretical results.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Bonnans, J.F., Jaisson, P.: Optimal control of a parabolic equation with time-dependent state constraints. SIAM J. Control Optim. 48, 4550–4571 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 3rd edn. Springer, Berlin (2007)

    Google Scholar 

  3. Casas, E.: Control of an elliptic problem with pointwise state constraints. SIAM J. Control Optim. 24, 1309–1318 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  4. Casas, E.: Boundary control of semilinear elliptic equations with pointwise state constraints. SIAM J. Control Optim. 31, 993–1006 (1993)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Ciarlet, P.G.: The Finite Element Methods for Elliptic Problems. North-Holland, Amsterdam (1978)

    Google Scholar 

  7. Deckelnick, K., Günther, A., Hinze, M.: Finite element approximation of elliptic control problems with constraints on the gradient. Numer. Math. 111, 335–350 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Deckelnick, K., Hinze, M.: Convergence of a fnite element approximation to a state constrained elliptic control problem. SIAM J. Numer. Anal. 45, 1937–1953 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Deckelnick, K., Hinze, M.: Variational discretization of parabolic control problems in the presence of pointwise state constraints. J. Comput. Math. 29, 1–16 (2011)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  11. Eriksson, K., Johnson, C.: Adaptive finite element methods for parabolic problems II: optimal error estimates in L ∞ L 2 and L ∞ L ∞. SIAM J. Numer. Anal. 32, 706–740 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  12. French, D.A., King, J.T.: Analysis of a robust finite element approximation for a parabolic equation with rough boundary data. Math. Comput. 60, 79–104 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gong, W., Yan, N.N.: A mixed finite element scheme for optimal control problems with pointwise state constraints. J. Sci. Comput. 46, 182–203 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hinze, M.: A variational discretization concept in control constrained optimization: the linear-quadratic case. Comput. Optim. Appl. 30, 45–63 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints MMTA 23. Springer, Berlin (2009)

    Google Scholar 

  16. Kahle, C.: Relaxierungsverfahren zur numerischen Lösung von parabolischen Kontrollproblemen mit Zustandsschranken. Diploma Thesis, Department Mathematik, Universits̈t Hamburg (2009)

  17. Ladyz̆enskaja, O.A., Solonnikov, V., Ural’ceva, N.: Linear and Quasilinear Equations of Parabolic Type. Translation Mathematical Monographs, vol. 23. AMS, Providence (1968)

    Google Scholar 

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

    Book  MATH  Google Scholar 

  19. Lions, J.L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications. Springer, Berlin (1972)

    Book  Google Scholar 

  20. Liu, W.B., Gong, W., Yan, N.N.: A new finite element approximation of a state-constrained optimal control problem. J. Comput. Math. 27, 97–114 (2009)

    MathSciNet  MATH  Google Scholar 

  21. Liu, W.B., Yan, N.N.: A posteriori error estimates for optimal control problems governed by parabolic equations. Numer. Math. 93, 497–521 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. Meidner, D., Rannacher, R., Vexler, B.: A priori error estimates for finite element discretizations of parabolic optimization problems with pointwise state constraints in time. SIAM J. Control Optim. 49, 1961–1997 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  24. Neitzel, I., Tröltzsch, F.: On regularization methods for the numerical solution of parabolic control problems with pointwise state constraints. ESAIM Control Optim. Calc. Var. 15, 426–453 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Nitsche, J.A.: L ∞-Convergence of finite element approximations. In: Mathematical Aspects of Finite Element Methods. Lecture Notes in Mathematics, vol. 606, pp. 261–274. Springer, Berlin (1977)

    Chapter  Google Scholar 

  26. Nochetto, R.H., Verdi, C.: Convergence past singularities for a fully discrete approximation of curvature-driven interfaces. SIAM J. Numer. Anal. 34, 490–512 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  27. Pierre, M., Sokolowsky, J.: Differentiability of Projection and Applications. Control of Partial Differential Equations and Applications (Laredo, 1994). Lecture Notes in Pure and Appl. Math., vol. 174, pp. 231–240. Dekker, New York (1996)

    Google Scholar 

  28. Rannacher, R., Scott, R.: Some optimal error estimates for piecewise linear finite element approximations. Math. Comput. 38, 437–445 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  29. Wang, G.S., Yu, X.: Error estimates for an optimal control problem governed by the heat equation with state and control constraints. Int. J. Numer. Anal. Model. 7, 30–65 (2010)

    MathSciNet  Google Scholar 

  30. Schatz, A.H.: Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids. I: Global estimates. Math. Comput. 67, 877–899 (1998)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Christian Kahle for the help on numerical experiments and Klaus Deckelnick for the discussion on Example 2. The first author also gratefully acknowledges the support of the Alexander von Humboldt Foundation during his stay at the University of Hamburg, Germany. He is also very grateful to the Department of Mathematics of the University of Hamburg for the hospitality and support. The first author was partially supported by the National Basic Research Program of China under the Grant 2012CB821204. The second author also gratefully acknowledges support of the priority program SPP1253 of the German research foundation.

Author information

Authors and Affiliations

  1. LSEC, Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Wei Gong

  2. Bereich Optimierung und Approximation, Universität Hamburg, Bundestrasse 55, 20146, Hamburg, Germany

    Wei Gong & Michael Hinze

Authors
  1. Wei Gong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michael Hinze
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wei Gong.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gong, W., Hinze, M. Error estimates for parabolic optimal control problems with control and state constraints. Comput Optim Appl 56, 131–151 (2013). https://doi.org/10.1007/s10589-013-9541-z

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-013-9541-z

Keywords

Search

Navigation