This paper presents a numerical fuzzy indirect method based on the fuzzy basis functions technique to solve an optimal control problem governed by Poisson’s differential equation. The considered problem may or may not be accompanied by a control box constraint. The first-order necessary optimality conditions have been derived, which may contain a variational inequality in function space. In the presented method, the obtained optimality conditions have been discretized using fuzzy basis functions and a system of equations introduced as the discretized optimality conditions. The derived system mostly contains some nonsmooth equations and conventional system solvers fail to solve them. A fuzzy system-based semi-smooth Newton method has also been introduced to deal with the obtained system. Solving optimality systems by the presented method gets us unknown fuzzy quantities on the state and control fuzzy expansions. Finally, some test problems have been studied to demonstrate the efficiency and accuracy of the presented fuzzy numerical technique.
[1] M. Azizi, M. Amirfakhrian, and M. A. Fariborzi Araghi, A fuzzy system based active set algorithm for the numerical solution of the optimal control problem governed by partial differential equation, Eur. J. Control, 54 (2020), 99–110.
[2] R. Becker, M. Braack, D. Meidner, R. Rannacher, and B. Vexler, Adaptive finite element methods for PDE- constrained optimal control problems, Reactive flows, diffusion and transport, Springer, 2007, 177–205.
[3] R. Becker, H. Kapp, and R. Rannacher, Adaptive finite element methods for optimal control of partial differential equations: Basic concept, SIAM J. Control Optim., 39(1) (2000), 113–132.
[4] M. Bergounioux, K. Ito, and K .Kunisch, Primal-dual strategy for constrained optimal control problems, SIAM J. Control Optim., 37(4) (1999), 1176–1194.
[5] M. Bergounioux and K. Kunisch, Augmented lagrangian techniques for elliptic state constrained optimal control problems, SIAM J. Control Optim., 35(5) (1997), 1524–1543.
[6] X. Chen, L. Qi and D. Sun, Global and superlinear convergence of the smoothing newton method and its application to general box constrained variational inequalities, Math. Comp., 67(222) (1998), 519–540.
[7] S. Effati and M. Pakdaman, Optimal control problem via neural networks, Neural Comput. & Applic., 23(7-8) (2013), 2093–2100.
[8] F. Facchinei and J. -S. Pang, Finite-dimensional variational inequalities and complementarity problems, Springer Science & Business Media, 2007.
[9] A. Fischer, Solution of monotone complementarity problems with locally lipschitzian functions, Math. Program- ming, 76(3) (1997), Ser. B, 513–532.
[10] S. Ghasemi and S. Effati, An artificial neural network for solving distributed optimal control of the Poisson’s equation, Neural Process Lett., 49 (2018), 159–175.
[11] C. Grossmann, H. -G. Roos, and M. Stynes, Numerical treatment of partial differential equations, vol. 154, Springer, 2007.
[12] M. Gugat and M. Herty, The smoothed-penalty algorithm for state constrained optimal control problems for partial differential equations, Optim. Methods Softw., 25(4-6) (2010), 573–599.
[13] M. Hintermüller and R. H. W. Hoppe, Goal-oriented adaptivity in control constrained optimal control of partial differential equations, SIAM J. Control Optim., 47(4) (2008), 1721–1743.
[14] M. Khaksar-e Oshagh and M. Shamsi, Direct pseudo-spectral method for optimal control of obstacle problem–an optimal control problem governed by elliptic variational inequality, Math. Methods Appl. Sci., 40(13) (2017), 4993–5004.
[15] M. Khaksar-e Oshagh and M. Shamsi, An adaptive wavelet collocation method for solving optimal control of elliptic variational inequalities of the obstacle type, Comput. Math. Appl., 75(2) (2018), 470–485.
[16] M. Khaksar-e Oshagh, M. Shamsi, and M. Dehghan, A wavelet-based adaptive mesh refinement method for the obstacle problem, Engineering with Computers, 34(3) (2018), 577–589.
[17] H. M. Kim and J. M. Mendel, Fuzzy basis functions: Comparisons with other basis functions, IEEE Trans. Fuzzy Syst., 3(2) (1995), 158–168.
[18] A. Kröner, K. Kunisch, and B. Vexler, Semismooth newton methods for optimal control of the wave equation with control constraints, SIAM J. Control Optim., 49(2) (2011), 830–858.
[19] R. Li, W. Liu, H. Ma, and T. Tang, Adaptive finite element approximation for distributed elliptic optimal control problems, SIAM J. Control Optim., 41(5) (2002), 1321–1349.
[20] W. Liu and N. Yan, A posteriori error estimates for distributed convex optimal control problems, Adv. Comput. Math., 15(1-4) (2001), 285–309.
[21] M. A. Mehrpouya and M. Khaksar-e Oshagh, An efficient numerical solution for time switching optimal control problems, Comput. Methods Differ. Equ., 9(1) (2021), 225–243.
[22] M. Pakdaman and S. Effati, Approximating the solution of optimal control problems by fuzzy systems, Neural Process Lett., 43(3) (2016), 667–686.
[23] J. W. Pearson, A radial basis function method for solving PDE-constrained optimization problems, Numer. Algor., 64(3) (2013), 481–506.
[24] J. W. Pearson and A. J. Wathen, A new approximation of the Schur complement in preconditioners for PDE- constrained optimization, Numer. Linear Algebra Appl., 19(5) (2012), 816–829.
[25] H. Pourbashash and M. Khaksar-e Oshagh, Local RBF-FD technique for solving the two-dimensional modified anomalous sub-diffusion equation, Appl. Math. Comput., 339 (2018), 144–152.
[26] L. Qi and J. Sun, A nonsmooth version of newton’s method, Math. Programming, 58(1) (1993), 353–367.
[27] F. Tröltzsch, Optimal control of partial differential equations: theory, methods, and applications, vol. 112, Amer- ican Mathematical Society, 2010.
[28] M. Ulbrich, Semismooth newton methods for operator equations in function spaces, SIAM J. Optim., 13(3) (2002), 805–841.
[29] L. X. Wang and J. M. Mendel, Back-propagation fuzzy system as nonlinear dynamic system identifiers, IEEE International Conference on Fuzzy Systems, 8 (1992), 1409–1418.
[30] L. X. Wang and J. M. Mendel, Fuzzy basis functions, universal approximation, and orthogonal least-squares learning, IEEE Transactions on Neural Networks, 3(5) (1992), 807–814.
Azizi, M., Amirfakhrian, M., & Fariborzi Araghi, M. A. (2022). Application of fuzzy systems on the numerical solution of the elliptic PDE-constrained optimal control problems. Computational Methods for Differential Equations, 10(2), 351-371. doi: 10.22034/cmde.2021.39351.1725
MLA
Masoomeh Azizi; Majid Amirfakhrian; Mohammad Ali Fariborzi Araghi. "Application of fuzzy systems on the numerical solution of the elliptic PDE-constrained optimal control problems". Computational Methods for Differential Equations, 10, 2, 2022, 351-371. doi: 10.22034/cmde.2021.39351.1725
HARVARD
Azizi, M., Amirfakhrian, M., Fariborzi Araghi, M. A. (2022). 'Application of fuzzy systems on the numerical solution of the elliptic PDE-constrained optimal control problems', Computational Methods for Differential Equations, 10(2), pp. 351-371. doi: 10.22034/cmde.2021.39351.1725
VANCOUVER
Azizi, M., Amirfakhrian, M., Fariborzi Araghi, M. A. Application of fuzzy systems on the numerical solution of the elliptic PDE-constrained optimal control problems. Computational Methods for Differential Equations, 2022; 10(2): 351-371. doi: 10.22034/cmde.2021.39351.1725
)
