Numerische Mathematik

, Volume 120, Issue 2, pp 345–386 | Cite as

A priori error estimates for space–time finite element discretization of semilinear parabolic optimal control problems

Article

Abstract

In this paper, a priori error estimates for space–time finite element discretizations of optimal control problems governed by semilinear parabolic PDEs and subject to pointwise control constraints are derived. We extend the approach from Meidner and Vexler (SIAM Control Optim 47(3):1150–1177, 2008; SIAM Control Optim 47(3):1301–1329, 2008) where linear-quadratic problems have been considered, discretizing the state equation by usual conforming finite elements in space and a discontinuous Galerkin method in time. Error estimates for controls discretized by piecewise constant functions in time and cellwise constant functions in space are derived in detail and we explain how error estimate for further discretization approaches, e.g., cellwise linear discretization in space, the postprocessing approach from Meyer and Rösch (SIAM J Control Optim 43:970–985, 2004), and the variationally discrete approach from Hinze (J Comput Optim Appl 30:45–63, 2005) can be obtained. In addition, we derive an estimate for a setting with finitely many time-dependent controls.

Mathematics Subject Classification (2000)

49M25 65M15 65M60 

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References

  1. 1.
    Arada N., Casas E., Tröltzsch F.: Error estimates for the numerical approximation of a semilinear elliptic control problem. Comput. Optim. Appl. 23, 201–229 (2002)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Casas E.: Pontryagin’s principle for state-constrained boundary control problems of semilinear parabolic equations. SIAM J. Control Optim. 35, 1297–1327 (1997)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Casas E., Mateos M.: Uniform convergence of the FEM. Applications to state constrained control problems. Comput. Appl. Math. 21(1), 67–100 (2002)MATHMathSciNetGoogle Scholar
  4. 4.
    Casas E., Mateos M., Tröltzsch F.: Error estimates for the numerical approximation of boundary semilinear elliptic control problems. Comput. Optim. Appl. 31, 193–220 (2005)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Casas E., Tröltzsch F.: Error estimates for the finite-element approximation of a semilinear elliptic control problem. Control Cybern. 31, 695–712 (2002)MATHGoogle Scholar
  6. 6.
    Chrysafinos K.: Convergence of discontinuous Galerkin approximations of an optimal control problem associated to semilinear parabolic PDE’s. ESAIM: M2AN 44(1), 189–206 (2010)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Ciarlet P.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)MATHGoogle Scholar
  8. 8.
    Clément P.: Approximation by finite element functions using local regularization. Revue Franc. Automat. Inform. Rech. Operat. 9(2), 77–84 (1975)Google Scholar
  9. 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)MathSciNetGoogle Scholar
  10. 10.
    Eriksson K., Estep D., Hansbo P., Johnson C.: Computational Differential Equations. Cambridge University Press, Cambridge (1996)MATHGoogle Scholar
  11. 11.
    Eriksson K., Johnson C., Thomée V.: Time discretization of parabolic problems by the discontinuous Galerkin method. M2AN Math. Model. Numer. Anal 19, 611–643 (1985)MATHGoogle Scholar
  12. 12.
    Evans L.C.: Partial Differential Equations. American Mathematical Society, Providence (1998)MATHGoogle Scholar
  13. 13.
    Falk R.S.: Approximation of a class of optimal control problems with order of convergence estimates. J. Math. Anal. Appl. 44, 28–47 (1973)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Gascoigne: The finite element toolkit. http://www.gascoigne.uni-hd.de/
  15. 15.
    Geveci T.: On the approximation of the solution of an optimal control problem problem governed by an elliptic equation. R.A.I.R.O. Analyse numérique/Numer. Anal. 13, 313–328 (1979)MATHMathSciNetGoogle Scholar
  16. 16.
    Hinze M.: A variational discretization concept in control constrained optimization: the linear-quadratic case. J. Comput. Optim. Appl. 30, 45–63 (2005)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Kröner A., Vexler B.: A priori error estimates for elliptic optimal control problems with bilinear state equation. J. Comput. Appl. Math. 230(2), 781–802 (2009)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Lasiecka I., Malanowski K.: On discrete-time Ritz-Galerkin approximation of control constrained optimal control problems for parabolic systems. Control Cybern. 7, 21–36 (1978)MATHMathSciNetGoogle Scholar
  19. 19.
    Malanowski K.: Convergence of approximations vs. regularity of solutions for convex, control–constrained optimal control problems. Appl. Math. Optim. 8, 69–95 (1981)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Mateos, M.: Problemas de Control Óptimo Gobernados por Ecuaciones Semilineales con Restricciones de Tipo Integral sobre el Gradiente del Estado. PhD thesis, Universidad de Cantabria (2000)Google Scholar
  21. 21.
    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 Control Optim. (2011, accepted)Google Scholar
  22. 22.
    Meidner D., Vexler B.: A priori error estimates for space–time finite element discretization of parabolic optimal control problems. Part I: problems without control constraints. SIAM Control Optim. 47(3), 1150–1177 (2008)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Meidner D., Vexler B.: A priori error estimates for space–time finite element discretization of parabolic optimal control problems. Part II: problems with control constraints. SIAM Control Optim. 47(3), 1301–1329 (2008)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Meyer C., Rösch A.: Superconvergence properties of optimal control problems. SIAM J. Control Optim. 43, 970–985 (2004)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    RoDoBo: AC++ library for optimization with stationary and nonstationary PDEs. http://www.rodobo.uni-hd.de/
  26. 26.
    Rösch A.: Error estimates for parabolic optimal control problems with control constraints. Z. für Analysis und ihre Anwendungen (ZAA) 23, 353–376 (2004)CrossRefMATHGoogle Scholar
  27. 27.
    Triebel H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam (1978)Google Scholar
  28. 28.
    Tröltzsch F.: Optimal Control of Partial Differential Equations: Theory, Methods and Applications. AMS, Providence (2010)MATHGoogle Scholar
  29. 29.
    Winther R.: Error estimates for a Galerkin approximation of a parabolic control problem. Ann. Math. Pura Appl. 117(4), 173–206 (1978)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Institut für MathematikTechnische Universität BerlinBerlinGermany
  2. 2.Zentrum MathematikTechnische Universität MünchenGarching b. MünchenGermany

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