10.1007/s40096-022-00488-8

Numerical solutions of two-dimensional PDE-constrained optimal control problems via bilinear pseudo-spectral method

XML
  • Fereshteh Samadi*1,
  • Aghileh Heydari1,
  • Sohrab Effati2,
  1. Department of Mathematics, Payame Noor University (PNU), Tehran, IR
  2. Department of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, IR

Published in Issue 2022-08-20

How to Cite

Samadi, F., Heydari, A., & Effati, S. (2022). Numerical solutions of two-dimensional PDE-constrained optimal control problems via bilinear pseudo-spectral method. Mathematical Sciences, 18(1 (March 2024). https://doi.org/10.1007/s40096-022-00488-8
Crossref
Scopus
Google Scholar
Europe PMC

Abstract

Abstract A bilinear pseudo-spectral method (BPSM) is proposed for solving two-dimensional parabolic optimal control problems (OCPs). Firstly, the OCP is converted to a partial differential equation system including the state equation of the main problem, the adjoint equation, and the gradient equation which should be solved. Secondly, the coupled system is discretized in the space domain by a BPSM based on Chebyshev polynomials and a coupled matrix differential equation is obtained. Lastly, by using the eigenvalue decomposition method, the equation is transformed to a set of independent coupled ordinary differential equation system in the new coordinate with opposite orientation and after another eigenvalue decomposition technique, these transformed equations are discretized in the time space by some higher-order formulas. The error estimate is investigated theoretically. Numerical results confirm the desirable accuracy and efficiency of the method.

Keywords

  • Parabolic optimal control problems,
  • Bilinear pseudo-spectral method,
  • Matrix differential equation,
  • Eigenvalue decomposition technique

References

  1. Abu-Rqayiqa et al. (2022) Optimal control of a basic model of oncolytic virotherapy (pp. 119-126) https://doi.org/10.22436/jmcs.024.02.03
  2. Becker and Vexler (2007) Optimal control of the convection-diffusion equation using stabilized finite element methods 106(3) (pp. 349-367) https://doi.org/10.1007/s00211-007-0067-0
  3. Borzı (2003) Multigrid methods for parabolic distributed optimal control problems 157(2) (pp. 365-382) https://doi.org/10.1016/S0377-0427(03)00417-5
  4. Borzì, A., Kunisch, K., Vanmaele, M.: A multigrid approach to the optimal control of solid fuel ignition problems. In: Multigrid Methods VI, pp. 59–65. Springer, Berlin, Heidelberg (2000)
  5. Burman (2011) Crank–Nicolson finite element methods using symmetric stabilization with an application to optimal control problems subject to transient advection-diffusion equations 9(1) (pp. 319-329) https://doi.org/10.4310/CMS.2011.v9.n1.a16
  6. Casanova, P.G., Zavaleta, J.: Radial basis function methods for optimal control of the convection-diffusion equation. arXiv preprint
  7. arXiv:1803.00915
  8. (2018)
  9. Chen et al. (2018) An HDG method for distributed control of convection diffusion PDEs (pp. 643-661) https://doi.org/10.1016/j.cam.2018.05.028
  10. Christofides and Chow (2002) Nonlinear and robust control of PDE systems: methods and applications to transport-reaction processes 55(2) (pp. B29-B30) https://doi.org/10.1115/1.1451164
  11. de Frutos et al. (2016) Local error estimates for the SUPG method applied to evolutionary convection-reaction-diffusion equations 66(2) (pp. 528-554) https://doi.org/10.1007/s10915-015-0035-2
  12. De los Reyes, J. C.: Numerical PDE-Constrained Optimization. Springer, Cham Heidelberg (2015)
  13. Dubiner (1987) Asymptotic analysis of spectral methods 2(1) (pp. 3-31) https://doi.org/10.1007/BF01061510
  14. Edmunds (1972) Optimal control of systems governed by partial differential equations 4(2) (pp. 236-237) https://doi.org/10.1112/blms/4.2.236
  15. Fu (2010) A characteristic finite element method for optimal control problems governed by convection-diffusion equations 235(3) (pp. 825-836) https://doi.org/10.1016/j.cam.2010.07.010
  16. Fu and Rui (2009) A priori error estimates for optimal control problems governed by transient advection-diffusion equations 38(3) (pp. 290-315) https://doi.org/10.1007/s10915-008-9224-6
  17. Ghasemi, S., Effati, S.: An artificial neural network for solving distributed optimal control of the Poisson’s equation. Neural Process. Lett.
  18. 49
  19. (1), 1–17 (2018)
  20. Hu et al. (2018) A superconvergent HDG method for distributed control of convection diffusion PDEs 76(3) (pp. 1436-1457) https://doi.org/10.1007/s10915-018-0668-z
  21. Kauffmann, A., Kunisch, K.: Optimal control of the solid fuel ignition model. In: ESAIM: Proceedings, EDP Sciences, vol. 8, pp. 65–76 (2000)
  22. Kunisch and Vexler (2007) Constrained Dirichlet boundary control in l2documentclass[12pt]{minimal}
  23. usepackage{amsmath}
  24. usepackage{wasysym}
  25. usepackage{amsfonts}
  26. usepackage{amssymb}
  27. usepackage{amsbsy}
  28. usepackage{mathrsfs}
  29. usepackage{upgreek}
  30. setlength{oddsidemargin}{-69pt}
  31. begin{document}$$^{2}$$end{document} for a class of evolution equations 46(5) (pp. 1726-1753) https://doi.org/10.1137/060670110
  32. Lapin et al. (2019) Numerical solution of a parabolic optimal control problem arising in economics and management (pp. 715-729)
  33. Lapin et al. (2019) Analysis of finite difference approximations of an optimal control problem in economics 11(6) (pp. 1358-1375) https://doi.org/10.4208/aamm.OA-2018-0186
  34. Lasiecka (1984) Ritz–Galerkin approximation of the time optimal boundary control problem for parabolic systems with Dirichlet boundary conditions 22(3) (pp. 477-500) https://doi.org/10.1137/0322029
  35. Li et al. (2017) A new multigrid method for unconstrained parabolic optimal control problems (pp. 358-373) https://doi.org/10.1016/j.cam.2017.06.008
  36. Liu and Wang (2018) Efficient time domain decomposition algorithms for parabolic PDE-constrained optimization problems 75(6) (pp. 2115-2133) https://doi.org/10.1016/j.camwa.2017.09.017
  37. Marin et al. (2020) A scalable matrix-free spectral element approach for unsteady PDE constrained optimization using PETSc/TAO https://doi.org/10.1016/j.jocs.2020.101207
  38. Martínez et al. (2000) Theoretical and numerical analysis of an optimal control problem related to wastewater treatment 38(5) (pp. 1534-1553) https://doi.org/10.1137/S0363012998345640
  39. Muzhinji, K.: Robust and efficient finite element multigrid and precondi-tioned minimum residual solvers for the distributed elliptic optimal control problems
  40. Ould Sidia et al. (2022) Regional averaged control problems with minimum energy constrained by distributed parabolic systems (pp. 349-356) https://doi.org/10.22436/jmcs.026.04.03
  41. Rashidinia et al. (2016) A stable method for the evaluation of Gaussian radial basis function solutions of interpolation and collocation problems 72(1) (pp. 178-193) https://doi.org/10.1016/j.camwa.2016.04.048
  42. Rashidinia and Khasi (2019) Stable Gaussian radial basis function method for solving Helmholtz equations 7(1) (pp. 138-151)
  43. Rashidinia et al. (2018) A stable Gaussian radial basis function method for solving nonlinear unsteady convection-diffusion-reaction equations 75(5) (pp. 1831-1850) https://doi.org/10.1016/j.camwa.2017.12.007
  44. Rezazadeh, A., Mahmoudi, M., Darehmiraki, M.: Space-time spectral collocation method for one-dimensional PDE constrained optimisation. Int. J. Control,
  45. 93
  46. (5), 1–11 (2018)
  47. Samadi, F., Heydari, A., Effati, S.: A numerical method based on a bilinear pseudo-spectral method to solve the convection-diffusion optimal control problems. Int. J. Comput. Math.
  48. 98
  49. (1), 1–20 (2020)
  50. Seymen et al. (2014) Distributed optimal control of time-dependent diffusion-convection-reaction equations using space-time discretization (pp. 146-157) https://doi.org/10.1016/j.cam.2013.11.006
  51. Shakya and Sinha (2020) Finite element method for parabolic optimal control problems with a bilinear state equation https://doi.org/10.1016/j.cam.2019.112431
  52. Shen (1994) Efficient spectral-Galerkin method I. Direct solvers of second-and fourth-order equations using Legendre polynomials 15(6) (pp. 489-1505) https://doi.org/10.1137/0915089
  53. Shen et al. (2011) Berlin Heidelberg
  54. Sidi (2020) Variational necessary conditions for optimal control problems 21(3) (pp. 186-191) https://doi.org/10.22436/jmcs.021.03.02
  55. Tröltzsch (2010) American Mathematical Society
  56. Wang et al. (2016) Optimal control of two-dimensional parabolic partial differential equations with application to steel billets cooling in continuous casting secondary cooling zone 37(6) (pp. 1314-1328) https://doi.org/10.1002/oca.2236
  57. Weideman and Trefethen (1988) The eigenvalues of second-order spectral differentiation matrices 25(6) (pp. 1279-1298) https://doi.org/10.1137/0725072
  58. Yücel et al. (2015) A discontinuous Galerkin method for optimal control problems governed by a system of convection-diffusion PDEs with nonlinear reaction terms 70(10) (pp. 2414-2431) https://doi.org/10.1016/j.camwa.2015.09.006
  59. Zhang et al. (2020) Immersed finite element method and its analysis for parabolic optimal control problems with interfaces (pp. 174-195) https://doi.org/10.1016/j.apnum.2019.08.024