Advertisement

A New Solution for Optimal Control of Fractional Convection–Reaction–Diffusion Equation Using Rational Barycentric Interpolation

  • Majid DarehmirakiEmail author
  • Arezou Rezazadeh
Original Paper
First Online:
  • 9 Downloads

Abstract

This paper solves an optimal control problem governed by a fractional convection–reaction–diffusion partial differential equation. Using Lagrangian multipliers, necessary conditions are obtained, and then, Barycentric collocation method are applied for discretizing classical derivatives and Grünwald–Letnikov formula for fractional derivative. Barycentric interpolation is a class of Lagrange polynomial interpolation that is fast and deserves to be known as a method of polynomial interpolation and Grünwald–Letnikov formula is a basic extension of the derivative in fractional calculus. Numerical examples are presented to show the effectiveness of the method.

Keywords

Optimal control Partial differential equation Convection–reaction fractional equation Grünwald–Letnikov formula Barycentric collocation method 

Mathematics Subject Classification

43A62 42C15 
This is a preview of subscription content, log in to check access.

Notes

References

  1. 1.
    Acosta, G., Borthagaray, J.P.: A fractional Laplace equation: regularity of solutions and finite element approximations. SIAM J. Numer. Anal. 55(2), 472–495 (2017)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Antil, H., Otarola, E.: A FEM for an optimal control problem of fractional powers of elliptic operators. SIAM J. Control Optim. 53(6), 3432–3456 (2015)MathSciNetCrossRefGoogle Scholar
  3. 3.
    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)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Berrut, J.P., Trefethen, L.N.: Barycentric lagrange interpolation. SIAM Rev. 46(3), 501–517 (2004)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Betts, J.T., Kolmanovsky, I.: Practical methods for optimal control using nonlinear programming. Appl. Mech. Rev. 55, B68 (2002)CrossRefGoogle Scholar
  6. 6.
    Biegler, L.T., Ghattas, O., Heinkenschloss, M., van Bloemen Waanders, B.: Large-scale PDE-constrained optimization: an introduction. In: Large-Scale PDE-Constrained Optimization (pp. 3–13). Springer, Berlin (2003)CrossRefGoogle Scholar
  7. 7.
    Bhrawy, A.H.: A new spectral algorithm for a time-space fractional partial differential equations with subdiffusion and superdiffusion. Proc. Rom. Acad. Ser. A 17(1), 39–47 (2016)MathSciNetGoogle Scholar
  8. 8.
    Bhrawy, A.H., Zaky, M.A., Van Gorder, R.A.: A space-time Legendre spectral tau method for the two-sided space-time Caputo fractional diffusion-wave equation. Numer. Algorithms 71(1), 151–180 (2016)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bonito, A., Pasciak, J.: Numerical approximation of fractional powers of elliptic operators. Math. Comput. 84(295), 2083–2110 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Borzì, A., Schulz, V.: Multigrid methods for PDE optimization. SIAM Rev. 51(2), 361–395 (2009)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Briggs, W.L., McCormick, S.F.: A Multigrid Tutorial, vol. 72. SIAM, Philadelphia (2000)CrossRefGoogle Scholar
  12. 12.
    Darehmiraki, M., Farahi, M.H., Effati, S.: A novel method to solve a class of distributed optimal control problems using Bezier curves. J. Comput. Nonlinear Dyn. 11(6), 061008 (2016)CrossRefGoogle Scholar
  13. 13.
    Darehmiraki, M., Farahi, M.H., Effati, S.: Solution for fractional distributed optimal control problem by hybrid meshless method. J. Vib. Control 24(11), 2149–2164 (2018)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Fix, G.J., Hackbusch, W.: Elliptic differential equations (theory and numerical treatment). Bull. Am. Math. Soc. 32(4), 458 (1995)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Floater, M.S., Hormann, K.: Barycentric rational interpolation with no poles and high rates of approximation. Numer. Math. 107(2), 315–331 (2007)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Gunzburger, M.D.: Perspectives in Flow Control and Optimization, vol. 5. SIAM, Philadelphia (2003)zbMATHGoogle Scholar
  17. 17.
    Hackbusch, W.: Multi-grid Methods and Applications, vol. 4. Springer Science and Business Media, Berlin (2013)zbMATHGoogle Scholar
  18. 18.
    Herzog, R., Kunisch, K.: Algorithms for PDE constrained optimization. GAMM-Mitteilungen 33(2), 163–176 (2010)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Hinze, M., Rösch, A.: Discretization of optimal control problems. In: Constrained Optimization and Optimal Control for Partial Differential Equations (pp. 391-430). Springer, Basel (2012)zbMATHGoogle Scholar
  20. 20.
    Johnson, C.: Numerical Solution of Partial Differential Equations by the Finite Element Method. Courier Corporation, Chelmsford (2012)Google Scholar
  21. 21.
    Klein, G., Berrut, J.P.: Linear rational finite differences from derivatives of Barycentric rational interpolants. SIAM J. Numer. Anal. 50(2), 643–656 (2012)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Leveque, R.J.: Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems, vol. 98. SIAM, Philadelphia (2007)CrossRefGoogle Scholar
  23. 23.
    Martínez, A., Rodríguez, C., Vázquez-Méndez, M.E.: Theoretical and numerical analysis of an optimal control problem related to wastewater treatment. SIAM J. Control Optim. 38, 1534–1553 (2000)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Mohammadzadeh, E., Pariz, N., Hosseini Sani, S.K., Jajarmi, A.: An efficient numerical method for the optimal control of fractional-order dynamic systems. J. Vib. Control. (2018).  https://doi.org/10.1177/1077546317751755 MathSciNetCrossRefGoogle Scholar
  25. 25.
    Mustapha, K., McLean, W.: Super convergence of a discontinuous Galerkin method for fractional diffusion and wave equations. SIAM J. Numer. Anal. 51(1), 491–515 (2013)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Nochetto, R.H., Otárola, E., Salgado, A.J.: A PDE approach to fractional diffusion in general domains: a priori error analysis. Found. Comput. Math. 15(3), 733–791 (2015)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Nochetto, R.H., Otarola, E., Salgado, A.J.: A PDE approach to space-time fractional parabolic problems. SIAM J. Numer. Anal. 54(2), 848–873 (2016)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Nocedal, J., Wright, S.J.: Numerical Optimization, 2nd edn. Springer Series in Operations Research and Financial Engineering. Springer, New York (2006)Google Scholar
  29. 29.
    Otarola, E.: A piecewise linear FEM for an optimal control problem of fractional operators: error analysis on curved domains. ESAIM Math. Model. Numer. Anal. 51(4), 1473–1500 (2017)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Otárola, E.: A PDE approach to numerical fractional diffusion (Doctoral dissertation). University of Maryland, College Park (2014)Google Scholar
  31. 31.
    Rezazadeh, A., Mahmoudi, M., Darehmiraki, M.: Space-time spectral collocation method for one-dimensional PDE constrained optimisation. Int. J. Control (2018).  https://doi.org/10.1080/00207179.2018.1501161 CrossRefGoogle Scholar
  32. 32.
    Saad, Y.: Iterative Methods for Sparse Linear Systems, vol. 82. SIAM, Philadelphia (2003)CrossRefGoogle Scholar
  33. 33.
    Scherer, R., Kalla, S.L., Tang, Y., Huang, J.: The Grünwald–Letnikov method for fractional differential equations. Comput. Math. Appl. 62(3), 902–917 (2011)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Steeb, W.H., Shi, T.K.: Matrix Calculus and Kronecker Product with Applications and C++ Programs. World Scientific, Singapore (1997)CrossRefGoogle Scholar
  35. 35.
    Steeb, W.H., Hardy, Y.: Problems and Solutions in Introductory and Advanced Matrix Calculus. World Scientific, Singapore (2016)CrossRefGoogle Scholar
  36. 36.
    Strikwerda, J.C.: Finite Difference Schemes and Partial Differential Equations, vol. 88. SIAM, Philadelphia (2004)zbMATHGoogle Scholar
  37. 37.
    Taylor, W.J.: Method of Lagrangian curvilinear interpolation. J. Res. Natl. Bureau Stand. 35(2), 151–155 (1945)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Van Loan, C.F., Golub, G.H.: Matrix Computations, p. 3. Johns Hopkins University Press, Baltimore (1983)zbMATHGoogle Scholar
  39. 39.
    Ye, X., Xu, C.: A spectral method for optimal control problems governed by the time fractional diffusion equation with control constraints. In: Spectral and High Order Methods for Partial Differential Equations-ICOSAHOM 2012, pp. 403–414. Springer, Cham (2014)Google Scholar
  40. 40.
    Yi, S.C., Yao, L.Q.: A steady Barycentric Lagrange interpolation method for the 2D higher-order time-fractional telegraph equation with nonlocal boundary condition with error analysis. Numer. Methods Partial Differ. Equ. (2019).  https://doi.org/10.1002/num.22371 MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Zhou, Z., Yu, X., Yan, N.: Local discontinuous galerkin approximation of convection-dominated diffusion optimal control problems with control constraints. Numer. Methods Partial Differ. Equ. 30(1), 339–360 (2014)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Zhu, J., Zeng, Q.C.: A mathematical formulation for optimal control of air pollution. Sci. China D 46, 994–1002 (2003)CrossRefGoogle Scholar
  43. 43.
    Zuppa, C.: Error estimates for moving least square approximations. Bull. Braz. Math. Soc. 34(2), 231–249 (2003)MathSciNetCrossRefGoogle Scholar

Copyright information

© Iranian Mathematical Society 2019

Authors and Affiliations

  1. 1.Department of MathematicsBehbahan Khatam Alanbia University of TechnologyBehbahanIran
  2. 2.Department of MathematicsUniversity of QomQomIran

Personalised recommendations

Cite article

Buy options