Skip to main content
Springer Nature Link
Log in

Finite Element Approximation of Optimal Control Problem Governed by Space Fractional Equation

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

In this paper we investigate finite element approximation of optimal control problem governed by space fractional diffusion equation with control constraints. The control variable is approximated by piecewise constant. Regularity estimate for the control problem is proved based on the first order optimality system and a priori error estimates for the state, the adjoint state and the control variables are derived. Due to the nonlocal property of fractional derivative, which will leads to a full stiff matrix, we develop a fast primal dual active set algorithm for the control problem. Numerical examples are given to illustrate the theoretical findings and the efficiency of the fast algorithm.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Benson, D.A., Wheatcraft, S.W., Meerschaeert, M.M.: The fractional order governing equations of Levy motion. Water Resour. Res. 36, 1413–1423 (2000)

    Article  Google Scholar 

  2. Meerschaert, M.M., Sikorskii, A.: Stochastic Models for Fractional Calculus, De Gruyter Studies in Mathematics, vol. 43. Walter de Gruyter, Berlin (2012)

    MATH  Google Scholar 

  3. Mophou, G.: Optimal control of fractional diffusion equation. Comput. Math. Appl. 61, 68–78 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Mophou, G., N’Guérékata, G.M.: Optimal control of fractional diffusion equation with state constraints. Comput. Math. Appl. 62, 1413–1426 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Fujishiro, K., Yamamoto, M.: Approximate controllability for fractional diffusion equations by interior control. Appl. Anal. 93(9), 1793–1810 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Sprekels, J., Valdinoci, E.: A new type of identification problems: optimizing the fractional order in a nonlocal evolution equation. SIAM J. Control. Optim. 55, 70–93 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ye, X.Y., Xu, C.J.: A spectral method for optimal control problem governed by the abnormal diffusion equation with integral constraint on the state. Sci. Sin. Math. 46, 1053–1070 (2016)

    Google Scholar 

  8. Ye, X.Y., Xu, C.J.: Spectral optimization methods for the time fractional diffusion inverse problem. Numer. Math. Theory Methods Appl. 6(3), 499–519 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ye, X.Y., Xu, C.J.: A space-time spectral method for the time fractional diffusion optimal control problems. Adv. Differ. Equ. 2015, 156 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Li, S.Y., Zhou, Z.J.: Legendre pseudo-spectral method for optimal control problem governed by a time-fractional diffusion equation. Int. J. Comput. Math. 95(6–7), 1308–1325 (2018)

    Article  MathSciNet  Google Scholar 

  11. Zaky, M.A., Machado, J.A.T.: On the formulation and numerical simulation of distributed-order fractional optimal control problems. Commun. Nonlinear Sci. Numer. Simul. 52, 177–189 (2017)

    Article  MathSciNet  Google Scholar 

  12. Antil, H., Otárola, E.: A FEM for an optimal control problem of fractional powers of elliptic operators. SIAM J. Control Optim. 53(6), 3432–3456 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Antil, H., Otárola, E., Salgado, A.J.: A space–time fractional optimal control problem: analysis and discretization. SIAM J. Control Optim. 54(3), 1295–1328 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Antil, H., Otárola, E.: An a posteriori error analysis for an optimal control problem involving the fractional Laplacian. IMA J. Numer. Anal. 38(1), 198–226 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  15. Antil, H., Otárola, E., Salgado, A.J.: Optimization with respect to order in a fractional diffusion model: analysis, approximation and algorithmic aspects. J. Sci. Comput. (2018). https://doi.org/10.1007/s10915-018-0703-0

    MathSciNet  MATH  Google Scholar 

  16. Biccari, U., Hernández-Santamaría, V.: Controllability of a one-dimensional fractional heat equation: theoretical and numerical aspects. IMA J. Math. Control Inform. (2018). https://doi.org/10.1093/imamci/dny025

    MATH  Google Scholar 

  17. Zhou, Z.J., Gong, W.: Finite element approximation of optimal control problems governed by time fractional diffusion equation. Comput. Math. Appl. 71, 301–318 (2016)

    Article  MathSciNet  Google Scholar 

  18. Jin, B.T., Li, B.Y., Zhou, Z.: Pointwise-in-time error estimates for an optimal control problem with subdiffusion constraint, arXiv:1707.08808

  19. Du, N., Wang, H., Liu, W.B.: A fast gradient projection method for a constrained fractional optimal control. J. Sci. Comput. 68, 1–20 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  20. Wang, H., Wang, K.X., Sircar, T.: A direct \(O(Nlog N)\) finite difference method for fractional diffusion equations. J. Comput. Phys. 229(21), 8095–8104 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ervin, V.J., Heuer, N., Roop, J.P.: Regularity of the solution to 1-D fractional order diffusion equations. Math. Comp. 87, 2273–2294 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kunisch, K., Vexler, B.: Constrained Dirichlet boundary control in \(L^2\) for a class of evolution equations. SIAM J. Control Optim. 46(5), 1726–1753 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Ervin, V.J., Roop, J.P.: Variational formulation for the stationary fractional advection dispersion equation. Numer. Methods Partial Differ. Equ. 22, 558–576 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Jin, B.T., Lazarov, R., Pasciak, J., Rundell, W.: Variational formulation of problems involving fractional order differential operators. Math. Comp. 84, 2665–2700 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  25. Bergounioux, M., Ito, K., Kunisch, K.: Primal–dual strategy for constrained optimal control problems. SIAM J. Control Optim. 37(4), 1176–1194 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  26. Li, Y.S., Chen, H.Z., Wang, H.: A mixed-type Galerkin variational formulation and fast algorithms for variable-coefficient fractional diffusion equations. Math. Methods Appl. Sci. 40(14), 5018–5034 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  27. Jia, L.L., Chen, H.Z., Wang, H.: Mixed-type Galerkin variational principle and numerical simulation for a generalized nonlocal elastic model. J. Sci. Comput. 71(2), 660–681 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  28. Davis, P.J.: Circulant Matrices. Wiley, New York (1979)

    MATH  Google Scholar 

  29. Gary, R.M.: Toeplitz and circulant matrices: a review. Found Trends Commun. Inf. Theory 2, 155–239 (2001)

    Article  Google Scholar 

  30. Gong, W., Hinze, M., Zhou, Z.J.: A priori error analysis for finite element approximation of parabolic optimal control problems with pointwise control. SIAM J. Control Optim. 52(1), 97–119 (2014)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the referees for their careful reviews and many valuable suggestions which have led to a considerably improved paper.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Shandong Normal University, Jinan, China

    Zhaojie Zhou

  2. Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong

    Zhiyu Tan

Authors
  1. Zhaojie Zhou
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Zhiyu Tan
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Zhiyu Tan.

Additional information

The research was supported by Natural Science Foundation of Shandong Province (No. ZR2016JL004, ZR2017MA020) and National Natural Science Foundation of China (No. 11301311, 11471196).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhou, Z., Tan, Z. Finite Element Approximation of Optimal Control Problem Governed by Space Fractional Equation. J Sci Comput 78, 1840–1861 (2019). https://doi.org/10.1007/s10915-018-0829-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-018-0829-0

Keywords