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

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.

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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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Berrut, J.P., Trefethen, L.N.: Barycentric lagrange interpolation. SIAM Rev. 46(3), 501–517 (2004)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Betts, J.T., Kolmanovsky, I.: Practical methods for optimal control using nonlinear programming. Appl. Mech. Rev. 55, B68 (2002)

    Article  Google 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)

  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)

    MathSciNet  Google 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)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Bonito, A., Pasciak, J.: Numerical approximation of fractional powers of elliptic operators. Math. Comput. 84(295), 2083–2110 (2015)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Borzì, A., Schulz, V.: Multigrid methods for PDE optimization. SIAM Rev. 51(2), 361–395 (2009)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Briggs, W.L., McCormick, S.F.: A Multigrid Tutorial, vol. 72. SIAM, Philadelphia (2000)

    Book  Google 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)

    Article  Google 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)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Fix, G.J., Hackbusch, W.: Elliptic differential equations (theory and numerical treatment). Bull. Am. Math. Soc. 32(4), 458 (1995)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Gunzburger, M.D.: Perspectives in Flow Control and Optimization, vol. 5. SIAM, Philadelphia (2003)

    MATH  Google Scholar 

  17. 17.

    Hackbusch, W.: Multi-grid Methods and Applications, vol. 4. Springer Science and Business Media, Berlin (2013)

    MATH  Google Scholar 

  18. 18.

    Herzog, R., Kunisch, K.: Algorithms for PDE constrained optimization. GAMM-Mitteilungen 33(2), 163–176 (2010)

    MathSciNet  Article  Google 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)

  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)

    MathSciNet  Article  Google 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)

    Book  Google 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)

    MathSciNet  Article  Google 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

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  MATH  Google Scholar 

  30. 30.

    Otárola, E.: A PDE approach to numerical fractional diffusion (Doctoral dissertation). University of Maryland, College Park (2014)

  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

    Article  MATH  Google Scholar 

  32. 32.

    Saad, Y.: Iterative Methods for Sparse Linear Systems, vol. 82. SIAM, Philadelphia (2003)

    Book  Google 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)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Steeb, W.H., Shi, T.K.: Matrix Calculus and Kronecker Product with Applications and C++ Programs. World Scientific, Singapore (1997)

    Book  Google Scholar 

  35. 35.

    Steeb, W.H., Hardy, Y.: Problems and Solutions in Introductory and Advanced Matrix Calculus. World Scientific, Singapore (2016)

    Book  Google Scholar 

  36. 36.

    Strikwerda, J.C.: Finite Difference Schemes and Partial Differential Equations, vol. 88. SIAM, Philadelphia (2004)

    MATH  Google Scholar 

  37. 37.

    Taylor, W.J.: Method of Lagrangian curvilinear interpolation. J. Res. Natl. Bureau Stand. 35(2), 151–155 (1945)

    MathSciNet  Article  Google Scholar 

  38. 38.

    Van Loan, C.F., Golub, G.H.: Matrix Computations, p. 3. Johns Hopkins University Press, Baltimore (1983)

    MATH  Google 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)

  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

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  Google Scholar 

  42. 42.

    Zhu, J., Zeng, Q.C.: A mathematical formulation for optimal control of air pollution. Sci. China D 46, 994–1002 (2003)

    Article  Google Scholar 

  43. 43.

    Zuppa, C.: Error estimates for moving least square approximations. Bull. Braz. Math. Soc. 34(2), 231–249 (2003)

    MathSciNet  Article  Google Scholar 

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Correspondence to Majid Darehmiraki.

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Darehmiraki, M., Rezazadeh, A. A New Solution for Optimal Control of Fractional Convection–Reaction–Diffusion Equation Using Rational Barycentric Interpolation. Bull. Iran. Math. Soc. 46, 1307–1340 (2020). https://doi.org/10.1007/s41980-019-00327-y

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Keywords

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

Mathematics Subject Classification

  • 43A62
  • 42C15