Skip to main content
SpringerLink
Log in

Analysis and Discretization for an Optimal Control Problem of a Variable-Coefficient Riesz-Fractional Diffusion Equation with Pointwise Control Constraints

  • Published:
Acta Mathematica Scientia Aims and scope Submit manuscript

Abstract

We present a mathematical and numerical study for a pointwise optimal control problem governed by a variable-coefficient Riesz-fractional diffusion equation. Due to the impact of the variable diffusivity coefficient, existing regularity results for their constant-coefficient counterparts do not apply, while the bilinear forms of the state (adjoint) equation may lose the coercivity that is critical in error estimates of the finite element method. We reformulate the state equation as an equivalent constant-coefficient fractional diffusion equation with the addition of a variable-coefficient low-order fractional advection term. First order optimality conditions are accordingly derived and the smoothing properties of the solutions are analyzed by, e.g., interpolation estimates. The weak coercivity of the resulting bilinear forms are proven via the Garding inequality, based on which we prove the optimal-order convergence estimates of the finite element method for the (adjoint) state variable and the control variable. Numerical experiments substantiate the theoretical predictions.

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

Similar content being viewed by others

References

  1. Acosta G, Borthagaray J P. A fractional Laplace equation: regularity of solutions and finite element approximations. SIAM J Numer Anal, 2017, 55: 472–495

    Article  MATH  Google Scholar 

  2. Adams R, Fournier J. Sobolev Spaces. San Diego: Elsevier, 2003

    MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  5. Benson D A, Wheatcraft S W, Meerschaeert M M. The fractional order governing equations of levy motion. Water Resour Res, 2000, 36(6): 1413–1423

    Article  Google Scholar 

  6. Brenner S C, Scott L R. The Mathematical Theory of Finite Element Methods. New York: Springer, 2008

    Book  MATH  Google Scholar 

  7. Becker R, Kapp H, Rannacher R. Adaptive finite element methods for optimal control of partial differential equations: Basic concept. SIAM J Control Optim, 2000, 39(1): 113–132

    Article  MATH  Google Scholar 

  8. Chen S, Shen J, Wang L L. Generalized Jacobi functions and their applications to fractional differential equations. Math Comp, 2016, 85: 1603–1638

    Article  MATH  Google Scholar 

  9. Chen S, Shen J. An efficient and accurate numerical method for the spectral fractional Laplacian equation. J Sci Comput, 2020, 82(1): 1–25

    Article  Google Scholar 

  10. D’Elia M, Glusa C, Otárola E. A priori error estimates for the optimal control of the integral fractional Laplacian. SIAM J Control Optim, 2019, 57: 2775–2798

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  12. Ervin V J, Roop J P. Variational formulation for the stationary fractional advection dispersion equation. Numer Meth for PDE, 2006, 22: 558–576

    Article  MATH  Google Scholar 

  13. Ervin V J. Regularity of the solution to fractional diffusion, advection, reaction equations in weighted Sobolev spaces. J Differ Equ, 2021, 278: 294–325

    Article  MATH  Google Scholar 

  14. Ervin V J, Heuer N, Roop J P. Regularity of the solution to 1-d fractional order diffusion equations. Math Comput, 2018, 87: 2273–2294

    Article  MATH  Google Scholar 

  15. Hao Z P, Park M, Lin G, et al. Finite element method for two-sided fractional differential equations with variable coefficients: Galerkin approach. J Sci Comput, 2019, 79: 700–717

    Article  MATH  Google Scholar 

  16. Hao Z P, Zhang Z Q. Optimal regularity and error estimates of a spectral Galerkin method for fractional advection-diffusion-reaction equations. SIAM J Numer Anal, 2020, 58(1): 211–233

    Article  MATH  Google Scholar 

  17. Kunisch K, Vexler B. Constrained Dirichlet boundary control in L2 for a class of evolution equations. SIAM J Control Optim, 2007, 46(5): 1726–1753

    Article  MATH  Google Scholar 

  18. Klafter J, Sokolov I M. Anomalous diffusion spreads its wings. Phys World, 2005, 18(8): 29–32

    Article  Google Scholar 

  19. Li S Y, Zhou Z J. Fractional spectral collocation method for optimal control problem governed by space fractional diffusion equation. Appl Math Comput, 2019, 350: 331–347

    MATH  Google Scholar 

  20. Ma L J, Liu B. Dynamic analysis and optimal control of a fractional order singular Leslie-Gower prey-predator model. Acta Math Sci, 2020, 40B(5): 1525–1552

    Article  MATH  Google Scholar 

  21. Mao Z P, Chen S, Shen J. Efficient and accurate spectral method using generalized Jacobi functions for solving Riesz fractional differential equations. Appl Numer Math, 2016, 106: 165–181

    Article  MATH  Google Scholar 

  22. Mao Z P, Karniadakis G E. A spectral method (of exponential convergence) for singular solutions of the diffusion equation with general two-sided fractional derivative. SIAM J Numer Anal, 2018, 56: 24–49

    Article  MATH  Google Scholar 

  23. Meerschaert M M, Sikorskii A. Stochastic Models for Fractional Calculus. Berlin: De Gruyter, 2019

    Book  MATH  Google Scholar 

  24. Nezza E D, Palatucci G, Valdinoci E. Hitchhiker’s guide to the fractional Sobolev spaces. Bulletin des Sciences Mathématiques, 2012, 136: 521–573

    Article  MATH  Google Scholar 

  25. Wang F Y, Zhang Z Q, Zhou Z J. A spectral Galerkin approximation of optimal control problem governed by fractional advection-diffusion-reaction equations. J Comput Appl Math, 2021, 386: 113233

    Article  MATH  Google Scholar 

  26. Wang H, Yang D P. Wellposedness of variable-coefficient conservative fractional elliptic differential equations. SIAM J Numer Anal, 2013, 51: 1088–1107

    Article  MATH  Google Scholar 

  27. Wang H, Yang D P, Zhu S F. Inhomogeneous Dirichlet boundary-value problems of space-fractional diffusion equations and their finite element approximations. SIAM J Numer Anal, 2014, 52: 1292–1310

    Article  MATH  Google Scholar 

  28. Zhang L, Zhou Z J. Spectral Galerkin approximation of optimal control problem governed by Riesz fractional differential equation. Appl Numer Math, 2019, 143: 247–262

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  30. Zaky M A. A Legendre collocation method for distributed-order fractional optimal control problems. Nonlinear Dyn, 2018, 91: 2667–2681

    Article  MATH  Google Scholar 

  31. Zhang Z Q. Error estimates of spectral Galerkin methods for a linear fractional reaction-diffusion equation. J Sci Comput, 2019, 78: 1087–1110

    Article  MATH  Google Scholar 

  32. Zheng X C, Ervin V J, Wang H. Numerical approximations for the variable coefficient fractional diffusion equations with non-smooth data. Comput Math Appl Math, 2020, 20(3): 573–589

    MATH  Google Scholar 

  33. Zheng X C, Ervin V J, Wang H. An indirect finite element method for variable-coefficient space-fractional diffusion equations and its optimal-order error estimates. Commun Appl Math Comput, 2020, 2: 147–162

    Article  MATH  Google Scholar 

  34. Zheng X C, Ervin V J, Wang H. Wellposedness of the two-sided variable coefficient Caputo flux fractional diffusion equation and error estimate of its spectral approximation. Appl Numer Math, 2020, 153: 234–237

    Article  MATH  Google Scholar 

  35. Zheng X C, Wang H. An optimal-order numerical approximation to variable-order space-fractional diffusion equations on uniform or graded meshes. SIAM J Numer Anal, 2020, 58(1): 330–352

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Shandong Normal University, Jinan, 250014, China

    Zhaojie Zhou  (周兆杰) & Fangyuan Wang  (王方圆)

  2. School of Mathematics, Shandong University, Jinan, 250100, China

    Xiangcheng Zheng  (郑祥成)

Authors
  1. Zhaojie Zhou  (周兆杰)
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fangyuan Wang  (王方圆)
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiangcheng Zheng  (郑祥成)
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiangcheng Zheng  (郑祥成).

Additional information

The first author was supported by the National Natural Science Foundation of China (11971276, 12171287), and by the Natural Science Foundation of Shandong Province (ZR2016JL004). The third author was supported by the China Postdoctoral Science Foundation (2021TQ0017, 2021M700244), and by the International Postdoctoral Exchange Fellowship Program (Talent-Introduction Program) (YJ20210019).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhou, Z., Wang, F. & Zheng, X. Analysis and Discretization for an Optimal Control Problem of a Variable-Coefficient Riesz-Fractional Diffusion Equation with Pointwise Control Constraints. Acta Math Sci 43, 640–654 (2023). https://doi.org/10.1007/s10473-023-0210-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10473-023-0210-8

Key words

2010 MR Subject Classification

Search

Navigation