A numerical method based on a bilinear pseudo-spectral method to solve the convection-diffusion optimal control problems

References

• V. Azhmyakov, M. Shamsi, L.A.G. Trujillo, and J.H.C. Rojas, A robust pseudo-spectral method for a class of optimal control problems, Automatica (Oxf) 48 (2015), pp. 253–258.
   Google Scholar
• R. Becker and B. Vexler, Optimal control of the convection-diffusion equation using stabilized finite element methods, Numer. Math. 16 (2007), pp. 349–367. doi: 10.1007/s00211-007-0067-0
   Web of Science ®Google Scholar
• R. Becker and B. Vexler, Optimal control in fluid mechanics by finite elements with symmetric stabilization, SIAM J. Control Optim. 48 (2009), pp. 672–687. doi: 10.1137/060653494
   Web of Science ®Google Scholar
• B.L. Buzbee, G.H. Golub, and C.W. Nielson, On direct methods for solving Poisson's equations, SIAM J. Numer. Anal. 7 (1970), pp. 627–656. doi: 10.1137/0707049
   Web of Science ®Google Scholar
• P.G. Casanova and J. Zavaleta, Radial basis function methods for optimal control of the convection-diffusion equation, preprint (2018). Available at arXiv:1803.00915.
   Google Scholar
• Y. Chen, N. Xia, and N. Yi, A Legendre Galerkin spectral method for optimal control problems, J. Syst. Sci. Complex. 24 (2011), pp. 663–671. doi: 10.1007/s11424-011-8016-5
   Web of Science ®Google Scholar
• G. Chen, W. Hu, J. Shen, J.R. Singler, and Y. Zhang, An HDG method for distributed control of convection diffusion PDEs, J. Comput. Appl. Math. 343 (2018), pp. 643–661. doi: 10.1016/j.cam.2018.05.028
   Web of Science ®Google Scholar
• S.S. Collis and M. Heinkenschloss, Analysis of the streamline upwind/Petrov Galerkin method applied to the solution of optimal control problems, CAAM TR02-01, 108, 2002.
   Google Scholar
• J. de Frutos, B. García-Archilla, and J. Novo, Local error estimates for the SUPG method applied to evolutionary convection–reaction–diffusion equations, J. Sci. Comput. 66 (2016), pp. 528–554. doi: 10.1007/s10915-015-0035-2
   Web of Science ®Google Scholar
• M. Dubiner, Asymptotic analysis of spectral methods, J. Sci. Comput. 2 (1987), pp. 3–31. doi: 10.1007/BF01061510
   Google Scholar
• S. Ghasemi, S. Effati, An artificial neural network for solving distributed optimal control of the Poisson’s equation. Neural Process. Lett. 49.1 (2019), pp. 159–175. doi: 10.1007/s11063-018-9806-8
   Web of Science ®Google Scholar
• M. Heinkenschloss and D. Leykekhman, Local error estimates for SUPG solutions of advection-dominated elliptic linear-quadratic optimal control problems, SIAM J. Numer. Anal. 47 (2010), pp. 4607–4638. doi: 10.1137/090759902
   Web of Science ®Google Scholar
• W. Hu, J. Shen, J.R. Singler, Y. Zhang, and X. Zheng, A superconvergent HDG method for distributed control of convection diffusion PDEs, J. Sci. Comput. 76 (2018), pp. 1436–1457. doi: 10.1007/s10915-018-0668-z
   Web of Science ®Google Scholar
• A. Martínez, C. Rodríguez, and M.E. Vázquez-Méndez, Theoretical and numerical analysis of an optimal control problem related to wastewater treatment, SIAM J. Control Optim. 38 (2000), pp. 1534–1553. doi: 10.1137/S0363012998345640
   Web of Science ®Google Scholar
• J.W. Pearson, A radial basis function method for solving PDE-constrained optimization problems, Numer. Algorithms 64 (2013), pp. 481–506. doi: 10.1007/s11075-012-9675-6
   Web of Science ®Google Scholar
• J. Rashidinia and M. Khasi, Stable Gaussian radial basis function method for solving Helmholtz equations, Comput. Methods Differ. Equ. 7 (2019), pp. 138–151.
   Web of Science ®Google Scholar
• J. Rashidinia, G.E. Fasshauer, and M. Khasi, A stable method for the evaluation of Gaussian radial basis function solutions of interpolation and collocation problems, Comput. Math. Appl. 72 (2016), pp. 178–193. doi: 10.1016/j.camwa.2016.04.048
   Web of Science ®Google Scholar
• J. Rashidinia, M. Khasi, and G.E. Fasshauer, A stable Gaussian radial basis function method for solving nonlinear unsteady convection–diffusion–reaction equations, Comput. Math. Appl. 75 (2018), pp. 1831–1850. doi: 10.1016/j.camwa.2017.12.007
   Web of Science ®Google Scholar
• T. Rees, Preconditioning iterative methods for PDE constrained optimization, Doctoral diss., University of Oxford, 2010.
   Google Scholar
• R. Salehi and M. Dehghan, The use of a Legendre pseudospectral viscosity technique to solve a class of nonlinear dynamic Hamilton–Jacobi equations, Comput. Math. Appl. 63 (2010), pp. 629–644. doi: 10.1016/j.camwa.2011.11.009
   Web of Science ®Google Scholar
• J. Shen, Efficient spectral-Galerkin method I. Direct solvers of second-and fourth-order equations using Legendre polynomials, SIAM J Sci. Comput. 15 (1994), pp. 1489–1505. doi: 10.1137/0915089
   Web of Science ®Google Scholar
• F. Tröltzsch, Optimal Control of Partial Differential Equations: Theory, Methods, and Applications, Vol. 112, American Mathematical Soc., United States of America, 2010.
   Google Scholar
• J. Shen, The eigenvalues of second-order spectral differentiation matrices, SIAM J Numer. Anal. 25 (1988), pp. 1279–1298. doi: 10.1137/0725072
   Web of Science ®Google Scholar
• H. Yücel, M. Stoll, and P. Benner, A discontinuous Galerkin method for optimal control problems governed by a system of convection–diffusion PDEs with nonlinear reaction terms, Comput. Math. Appl. 70 (2015), pp. 2414–2431. doi: 10.1016/j.camwa.2015.09.006
   Web of Science ®Google Scholar
• J. Zhu and Q. Zeng, A mathematical formulation for optimal control of air pollution, Sci. China Ser. D Earth Sci. 46 (2003), pp. 994–1002. doi: 10.1007/BF02959394
   Google Scholar