Advertisement

Journal of Systems Science and Complexity

, Volume 26, Issue 6, pp 902–924 | Cite as

Variational discretization for optimal control problems governed by parabolic equations

  • Yanping ChenEmail author
  • Tianliang Hou
  • Nianyu Yi
Article
  • 165 Downloads

Abstract

This paper considers the variational discretization for the constrained optimal control problem governed by linear parabolic equations. The state and co-state are approximated by Raviart-Thomas mixed finite element spaces, and the authors do not discretize the space of admissible control but implicitly utilize the relation between co-state and control for the discretization of the control. A priori error estimates are derived for the state, the co-state, and the control. Some numerical examples are presented to confirm the theoretical investigations.

Key words

A priori error estimates mixed finite element methods optimal control problems parabolic equations variational discretization 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Ciarlet P G, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.zbMATHGoogle Scholar
  2. [2]
    Haslinger J and Neittaanmaki P, Finite Element Approximation for Optimal Shape Design, John Wiley and Sons, Chichester, UK, 1989.Google Scholar
  3. [3]
    Hinze M, Pinnau R, Ulbrich M, and Ulbrich S, Optimization with PDE Constraints, Springer-Verlag, Berlin, 2009.zbMATHGoogle Scholar
  4. [4]
    Lions J L, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, 1971.CrossRefzbMATHGoogle Scholar
  5. [5]
    Liu W B and Yan N, Adaptive Finite Element Methods for Optimal Control Governed by PDEs, Science Press, Beijing, 2008.Google Scholar
  6. [6]
    Neittaanmaki P and Tiba D, Optimal Control of Nonlinear Parabolic Systems: Theroy, Algorithms, and Applications, Dekker M, New York, 1994.Google Scholar
  7. [7]
    Tiba D, Lectures on the optimal control of elliptic equations, University of Jyvaskyla Press, Finland, 1995.Google Scholar
  8. [8]
    Alt W, On the approximation of infinite optimization problems with an application to optimal control problems, Appl. Math. Optim., 1984, 12(1): 15–27.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Alt W and Machenroth U, Convergence of finite element approximations to state constrained convex parabolic boundary control problems, SIAM J. Control Optim., 1989, 27(4): 718–736.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Arada N, Casas E, and Tröltzsch F, Error estimates for the numerical approximation of a semilinear elliptic control problem, Comput. Optim. Appl., 2002, 23(2): 201–229.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Casas E, Optimal control in coefficients of elliptic equations with state constraints, Appl. Math. Optim, 1992, 26(1): 21–37.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Falk F S, Approximation of a class of optimal control problems with order of convergence estimates, J. Math. Anal. Appl, 1973, 44(1): 28–47.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Geveci T, On the approximation of the solution of an optimal control problem governed by an elliptic equation, RAIRO. Anal. Numer., 1979, 13: 313–328.MathSciNetzbMATHGoogle Scholar
  14. [14]
    Hinze M, A variational discretization concept in control constrained optimization: The linearquadratic case, Comput. Optim. Appl., 2005, 30: 45–63.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Tiba D and Troltzsch F, Error estimates for the discretization of state constrained convex control problems, Numer. Funct. Anal. Optim., 1996, 17(9–10): 1005–1028.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Chen Y P and Dai Y Q, Superconvergence for optimal control problems governed by semi-linear elliptic equations, J. Sci. Comput., 2009, 39(2): 206–221.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    Meyer C and Rösch A, Superconvergence properties of optimal control problems, SIAM J. Control Optim., 2004, 43(3): 970–985.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    Becker R, Kapp H, and Rancher R, Adaptive finite element methods for optimal control of partial differential equations: Basic concept, SIAM J. Control Optim., 2000, 39(1): 113–132.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    Ge L, Liu W B, and Yang D P, Adaptive finite element approximation for a constrained optimal control problem via multi-meshes, J. Sci. Comput, 2009, 41(2): 238–255.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    Huang Y, Li R, Liu W B, and Yan N, Efficient discretization for finite element approximation of constrained optimal control problems, submitted.Google Scholar
  21. [21]
    Li R, Liu W B, Ma H, and Tang T, Adaptive finite element approximation of elliptic optimal control, SIAM J. Control Optim., 2002, 41(5): 1321–1349.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    Liu W B and Tiba D, Error estimates for the finite element approximation of nonlinear optimal control problems, J. Numer. Func. Optim., 2001, 22: 953–972.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    Liu W B and Yan N, A posteriori error analysis for convex distributed optimal control problems, Adv. Comp. Math., 2001, 15(1–4): 285–309.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    Knowles G, Finite element approximation of parabolic time optimal control problems, SIAM J. Control Optim., 1982, 20(3): 414–427.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    Liu W B and Yan N, A posteriori error estimates for optimal control problems governed by parabolic equations, Numer. Math., 2003, 93(3): 497–521.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    McKinght R S and Borsarge J, The Rite-Galerkin procedure for parabolic control problems, SIAM J. Control Optim., 1973, 11(3): 510–542.CrossRefGoogle Scholar
  27. [27]
    Chen Y, Superconvergence of mixed finite element methods for optimal control problems, Math. Comp., 2008, 77(263): 1269–1291.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    Chen Y, Superconvergence of quadratic optimal control problems by triangular mixed finite elements, Inter. J. Numer. Meths. Eng., 2008, 75(8): 881–898.CrossRefzbMATHGoogle Scholar
  29. [29]
    Chen Y, Huang Y, Liu W B, and Yan N, Error estimates and superconvergence of mixed finite element methods for convex optimal control problems, J. Sci. Comput., 2009, 42(3): 382–403.MathSciNetCrossRefGoogle Scholar
  30. [30]
    Chen Y, Yi N, and Liu W B, A Legendre Galerkin spectral method for optimal control problems governed by elliptic equations, SIAM J. Numer. Anal., 2008, 46(5): 2254–2275.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    Lions J L and Magenes E, Non Homogeneous Boundary Value Problems and Applications, Springer-Verlag, Berlin, 1972.CrossRefGoogle Scholar
  32. [32]
    Douglas J and Roberts J E, Global estimates for mixed finite element methods for second order elliptic equations, Math. Comp., 1985, 44(169): 39–52.MathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    Raviart P A and Thomas J M, A mixed finite element method for 2nd order elliptic problems, aspects of the finite element method, Lecture Notes in Math., Springer-Verlag, Berlin, 1977, 606: 292–315.Google Scholar
  34. [34]
    Brezzi F and Fortin M, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991.CrossRefzbMATHGoogle Scholar
  35. [35]
    Garcia S F M, Improved error estimates for mixed finite element approximations for nonlinear parabolic equations: The continuous-time case, Numer. Methods Partial Differential Eq., 1994, 10: 129–147.CrossRefzbMATHGoogle Scholar
  36. [36]

Copyright information

© Institute of Systems Science, Academy of Mathematics and Systems Science, CAS and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.School of Mathematical SciencesSouth China Normal UniversityGuangzhouChina
  2. 2.School of Mathematics and Computational ScienceXiangtan UniversityXiangtanChina

Personalised recommendations