August  2022, 27(8): 4121-4141. doi: 10.3934/dcdsb.2021220

Galerkin spectral method for elliptic optimal control problem with $L^2$-norm control constraint

1. 

School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China

2. 

Beijing Key Laboratory on MCAACI, Beijing Institute of Technology, Beijing 100081, China

*Corresponding author: Bing Sun

Received  March 2021 Revised  July 2021 Published  August 2022 Early access  September 2021

Fund Project: The second author is supported in part by the National Natural Science Foundation of China under Grant No. 11471036

This paper is concerned with the Galerkin spectral approximation of an optimal control problem governed by the elliptic partial differential equations (PDEs). Its objective functional depends on the control variable governed by the $ L^2 $-norm constraint. The optimality conditions for both the optimal control problem and its corresponding spectral approximation problem are given, successively. Thanks to some lemmas and the auxiliary systems, a priori error estimates of the Galerkin spectral approximation problem are established in detail. Moreover, a posteriori error estimates of the spectral approximation problem are also investigated, which include not only $ H^1 $-norm error for the state and co-state but also $ L^2 $-norm error for the control, state and costate. Finally, three numerical examples are executed to demonstrate the errors decay exponentially fast.

Citation: Zhen-Zhen Tao, Bing Sun. Galerkin spectral method for elliptic optimal control problem with $L^2$-norm control constraint. Discrete and Continuous Dynamical Systems - B, 2022, 27 (8) : 4121-4141. doi: 10.3934/dcdsb.2021220
References:
[1]

B. L. BuzbeeG. H. Golub and C. W. Nielson, On direct methods for solving Poisson's equations, SIAM J. Numer. Anal., 7 (1970), 627-656.  doi: 10.1137/0707049.

[2]

C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer-Verlag, Berlin, 1988. doi: 10.1007/978-3-642-84108-8.

[3]

C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods, Fundamentals in Single Domains, Scientific Computation, Springer-Verlag, Berlin, 2006. doi: 10.1007/978-3-540-30726-6.

[4]

E. Casas, Control of an elliptic problem with pointwise state constraints, SIAM J. Control Optim., 24 (1986), 1309-1318.  doi: 10.1137/0324078.

[5]

E. CasasM. Mateos and A. Rösch, Improved approximation rates for a parabolic control problem with an objective promoting directional sparsity, Comput. Optim. Appl., 70 (2018), 239-266.  doi: 10.1007/s10589-018-9979-0.

[6]

E. CasasM. Mateos and A. Rösch, Error estimates for semilinear parabolic control problems in the absence of Tikhonov term, SIAM J. Control Optim., 57 (2019), 2515-2540.  doi: 10.1137/18M117220X.

[7]

Y. P. Chen and F. L. Huang, Galerkin spectral approximation of elliptic optimal control problems with $H^1$-norm state constraint, J. Sci. Comput., 67 (2016), 65-83.  doi: 10.1007/s10915-015-0071-y.

[8]

Y. P. Chen and F. L. Huang, Spectral method approximation of flow optimal control problems with $H^1$-norm state constraint, Numer. Math. Theory Methods Appl., 10 (2017), 614-638.  doi: 10.4208/nmtma.2017.m1419.

[9]

Y. P. ChenF. L. HuangN. Y. Yi and W. B. Liu, A Legendre-Galerkin spectral method for optimal control problems governed by Stokes equations, SIAM J. Numer. Anal., 49 (2011), 1625-1648.  doi: 10.1137/080726057.

[10]

Y. P. ChenN. S. Xia and N. Y. Yi, A Legendre Galerkin spectral method for optimal control problems, J. Syst. Sci. Complex., 24 (2011), 663-671.  doi: 10.1007/s11424-011-8016-5.

[11]

Y. P. ChenN. Y. Yi and W. B. Liu, A Legendre-Galerkin spectral method for optimal control problems governed by elliptic equations, SIAM J. Numer. Anal., 46 (2008), 2254-2275.  doi: 10.1137/070679703.

[12]

K. Deckelnick and M. Hinze, Convergence of a finite element approximation to a state-constrained elliptic control problem, SIAM J. Numer. Anal., 45 (2007), 1937-1953.  doi: 10.1137/060652361.

[13]

D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 26, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1977. doi: 10.1137/1.9781611970425.

[14]

B. Y. Guo, Spectral Methods and Their Applications, World Scientific, Singapore, 1998. doi: 10.1142/3662.

[15]

B. Y. Guo, Error estimation of Hermite spectral method for nonlinear partial differential equations, Math. Comput., 68 (1999), 1067-1078.  doi: 10.1090/S0025-5718-99-01059-5.

[16]

M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints, Mathematical Modelling: Theory and Applications, Vol 23, Springer, New York, 2009. doi: 10.1007/978-1-4020-8839-1.

[17]

F. L. Huang and Y. P. Chen, Error estimates for spectral approximation of elliptic control problems with integral state and control constraints, Comput. Math. Appl., 68 (2014), 789-803.  doi: 10.1016/j.camwa.2014.07.002.

[18]

F. L. HuangZ. Zheng and Y. C. Peng, Error estimates of the space-time spectral method for parabolic control problems, Comput. Math. Appl., 75 (2018), 335-348.  doi: 10.1016/j.camwa.2017.09.018.

[19]

X. X. LinY. P. Chen and Y. Q. Huang, Galerkin spectral approximation of optimal control problems with $L^2$-norm control constraint, Appl. Numer. Math., 150 (2020), 418-432.  doi: 10.1016/j.apnum.2019.10.014.

[20]

J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, 1971.

[21]

W. B. Liu and N. N. Yan, A posteriori error estimates for control problems governed by Stokes equations, SIAM J. Numer. Anal., 40 (2002), 1850-1869.  doi: 10.1137/S0036142901384009.

[22] W. B. Liu and N. N. Yan, Adaptive Finite Element Methods for Optimal Control Governed by PDEs, Science Press, Beijing, 2008. 
[23]

W. B. LiuD. P. YangL. Yuan and C. Ma, Finite element approximations of an optimal control problem with integral state constraint, SIAM J. Numer. Anal., 48 (2010), 1163-1185.  doi: 10.1137/080737095.

[24]

I. NeitzelJ. Pfefferer and A. Rösch, Finite element discretization of state-constrained elliptic optimal control problems with semilinear state equation, SIAM J. Control Optim., 53 (2015), 874-904.  doi: 10.1137/140960645.

[25]

J. Shen, Efficient Spectral-Galerkin method I: Direct solvers for second and fourth order equations using Legendre polynomials, SIAM J. Sci. Comput., 15 (1994), 1489-1505.  doi: 10.1137/0915089.

[26]

J. Shen, On fast direct Possion solver, inf-sup constant and iterative Stokes solver by Legendre-Galerkin method, J. Comput. Phys., 116 (1995), 184-188.  doi: 10.1006/jcph.1995.1017.

[27] J. Shen and T. Tang, Spectral and High-Order Methods with Applications, Mathematics Monograph Series, Vol. 3, Science Press, Beijing, 2006. 
[28]

L. N. Trefethen, Spectral Methods in MATLAB, Software, Environments, and Tools, Vol. 10, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2000. doi: 10.1137/1.9780898719598.

[29]

F. Tröltzsch, Optimal Control of Partial Differential Equations: Theory, Methods and Applications, Graduate Studies in Mathematics, Vol. 112, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/112.

[30]

L. X. Zhou, A priori error estimates for optimal control problems with state and control constraints, Optim. Control Appl. Meth., 39 (2018), 1168-1181.  doi: 10.1002/oca.2402.

[31]

J. W. Zhou and D. P. Yang, Spectral mixed Galerkin method for state constrained optimal control problem governed by the first bi-harmonic equation, Int. J. Comput. Math., 88 (2011), 2988-3011.  doi: 10.1080/00207160.2011.563845.

[32]

J. W. Zhou and D. P. Yang, Legendre-Galerkin spectral methods for optimal control problems with integral constraint for state in one dimension, Comput. Optim. Appl., 61 (2015), 135-158.  doi: 10.1007/s10589-014-9700-x.

[33]

J. W. ZhouJ. Zhang and X. Q. Xing, Galerkin spectral approximations for optimal control problems governed by the fourth order equation with an integral constraint on state, Comput. Math. Appl., 72 (2016), 2549-2561.  doi: 10.1016/j.camwa.2016.08.009.

show all references

References:
[1]

B. L. BuzbeeG. H. Golub and C. W. Nielson, On direct methods for solving Poisson's equations, SIAM J. Numer. Anal., 7 (1970), 627-656.  doi: 10.1137/0707049.

[2]

C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer-Verlag, Berlin, 1988. doi: 10.1007/978-3-642-84108-8.

[3]

C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods, Fundamentals in Single Domains, Scientific Computation, Springer-Verlag, Berlin, 2006. doi: 10.1007/978-3-540-30726-6.

[4]

E. Casas, Control of an elliptic problem with pointwise state constraints, SIAM J. Control Optim., 24 (1986), 1309-1318.  doi: 10.1137/0324078.

[5]

E. CasasM. Mateos and A. Rösch, Improved approximation rates for a parabolic control problem with an objective promoting directional sparsity, Comput. Optim. Appl., 70 (2018), 239-266.  doi: 10.1007/s10589-018-9979-0.

[6]

E. CasasM. Mateos and A. Rösch, Error estimates for semilinear parabolic control problems in the absence of Tikhonov term, SIAM J. Control Optim., 57 (2019), 2515-2540.  doi: 10.1137/18M117220X.

[7]

Y. P. Chen and F. L. Huang, Galerkin spectral approximation of elliptic optimal control problems with $H^1$-norm state constraint, J. Sci. Comput., 67 (2016), 65-83.  doi: 10.1007/s10915-015-0071-y.

[8]

Y. P. Chen and F. L. Huang, Spectral method approximation of flow optimal control problems with $H^1$-norm state constraint, Numer. Math. Theory Methods Appl., 10 (2017), 614-638.  doi: 10.4208/nmtma.2017.m1419.

[9]

Y. P. ChenF. L. HuangN. Y. Yi and W. B. Liu, A Legendre-Galerkin spectral method for optimal control problems governed by Stokes equations, SIAM J. Numer. Anal., 49 (2011), 1625-1648.  doi: 10.1137/080726057.

[10]

Y. P. ChenN. S. Xia and N. Y. Yi, A Legendre Galerkin spectral method for optimal control problems, J. Syst. Sci. Complex., 24 (2011), 663-671.  doi: 10.1007/s11424-011-8016-5.

[11]

Y. P. ChenN. Y. Yi and W. B. Liu, A Legendre-Galerkin spectral method for optimal control problems governed by elliptic equations, SIAM J. Numer. Anal., 46 (2008), 2254-2275.  doi: 10.1137/070679703.

[12]

K. Deckelnick and M. Hinze, Convergence of a finite element approximation to a state-constrained elliptic control problem, SIAM J. Numer. Anal., 45 (2007), 1937-1953.  doi: 10.1137/060652361.

[13]

D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 26, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1977. doi: 10.1137/1.9781611970425.

[14]

B. Y. Guo, Spectral Methods and Their Applications, World Scientific, Singapore, 1998. doi: 10.1142/3662.

[15]

B. Y. Guo, Error estimation of Hermite spectral method for nonlinear partial differential equations, Math. Comput., 68 (1999), 1067-1078.  doi: 10.1090/S0025-5718-99-01059-5.

[16]

M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints, Mathematical Modelling: Theory and Applications, Vol 23, Springer, New York, 2009. doi: 10.1007/978-1-4020-8839-1.

[17]

F. L. Huang and Y. P. Chen, Error estimates for spectral approximation of elliptic control problems with integral state and control constraints, Comput. Math. Appl., 68 (2014), 789-803.  doi: 10.1016/j.camwa.2014.07.002.

[18]

F. L. HuangZ. Zheng and Y. C. Peng, Error estimates of the space-time spectral method for parabolic control problems, Comput. Math. Appl., 75 (2018), 335-348.  doi: 10.1016/j.camwa.2017.09.018.

[19]

X. X. LinY. P. Chen and Y. Q. Huang, Galerkin spectral approximation of optimal control problems with $L^2$-norm control constraint, Appl. Numer. Math., 150 (2020), 418-432.  doi: 10.1016/j.apnum.2019.10.014.

[20]

J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, 1971.

[21]

W. B. Liu and N. N. Yan, A posteriori error estimates for control problems governed by Stokes equations, SIAM J. Numer. Anal., 40 (2002), 1850-1869.  doi: 10.1137/S0036142901384009.

[22] W. B. Liu and N. N. Yan, Adaptive Finite Element Methods for Optimal Control Governed by PDEs, Science Press, Beijing, 2008. 
[23]

W. B. LiuD. P. YangL. Yuan and C. Ma, Finite element approximations of an optimal control problem with integral state constraint, SIAM J. Numer. Anal., 48 (2010), 1163-1185.  doi: 10.1137/080737095.

[24]

I. NeitzelJ. Pfefferer and A. Rösch, Finite element discretization of state-constrained elliptic optimal control problems with semilinear state equation, SIAM J. Control Optim., 53 (2015), 874-904.  doi: 10.1137/140960645.

[25]

J. Shen, Efficient Spectral-Galerkin method I: Direct solvers for second and fourth order equations using Legendre polynomials, SIAM J. Sci. Comput., 15 (1994), 1489-1505.  doi: 10.1137/0915089.

[26]

J. Shen, On fast direct Possion solver, inf-sup constant and iterative Stokes solver by Legendre-Galerkin method, J. Comput. Phys., 116 (1995), 184-188.  doi: 10.1006/jcph.1995.1017.

[27] J. Shen and T. Tang, Spectral and High-Order Methods with Applications, Mathematics Monograph Series, Vol. 3, Science Press, Beijing, 2006. 
[28]

L. N. Trefethen, Spectral Methods in MATLAB, Software, Environments, and Tools, Vol. 10, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2000. doi: 10.1137/1.9780898719598.

[29]

F. Tröltzsch, Optimal Control of Partial Differential Equations: Theory, Methods and Applications, Graduate Studies in Mathematics, Vol. 112, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/112.

[30]

L. X. Zhou, A priori error estimates for optimal control problems with state and control constraints, Optim. Control Appl. Meth., 39 (2018), 1168-1181.  doi: 10.1002/oca.2402.

[31]

J. W. Zhou and D. P. Yang, Spectral mixed Galerkin method for state constrained optimal control problem governed by the first bi-harmonic equation, Int. J. Comput. Math., 88 (2011), 2988-3011.  doi: 10.1080/00207160.2011.563845.

[32]

J. W. Zhou and D. P. Yang, Legendre-Galerkin spectral methods for optimal control problems with integral constraint for state in one dimension, Comput. Optim. Appl., 61 (2015), 135-158.  doi: 10.1007/s10589-014-9700-x.

[33]

J. W. ZhouJ. Zhang and X. Q. Xing, Galerkin spectral approximations for optimal control problems governed by the fourth order equation with an integral constraint on state, Comput. Math. Appl., 72 (2016), 2549-2561.  doi: 10.1016/j.camwa.2016.08.009.

Figure 1.  Error estimates of Example 5.1
Figure 2.  The point-wise results of $ u-u_N $, $ y-y_N $ and $ p-p_N $ for Example 5.1
Figure 3.  Error estimates of Example 5.2
Figure 4.  The exact solutions and its spectral solutions of Example 5.2 when $ N = 28 $
Figure 5.  Error estimates of Example 5.3
Figure 6.  The pointwise result of $ u-u_N $ for Example 5.3
Figure 7.  The pointwise result of $ y-y_N $ for Example 5.3
Table 1.  Error results between the numerical solutions and the exact solutions of Example 5.1
$ N $ 8 12 16 20 24
$ \|u-u_N\|_{L^2(\Omega)} $ 3.2745e-4 1.2301e-7 1.3536e-11 7.5344e-16 3.6242e-16
$ \|y-y_N\|_{L^2(\Omega)} $ 2.0412e-4 1.6475e-7 3.1550e-11 2.1122e-15 3.0444e-17
$ \|p-p_N\|_{L^2(\Omega)} $ 1.6373e-4 6.1497e-8 6.7677e-12 3.3969e-16 1.1459e-16
$ \|y-y_N\|_{H^1(\Omega)} $ 2.3942e-3 2.8825e-6 7.3202e-10 6.0843e-14 1.1835e-16
$ \|p-p_N\|_{H^1(\Omega)} $ 1.9464e-3 1.0822e-6 1.5764e-10 8.5485e-15 3.9737e-16
$ \Sigma_1 $ 4.6681e-3 4.0878e-6 9.0320e-10 7.0145e-14 8.7814e-16
$ \Sigma_2 $ 6.9530e-4 3.4925e-7 5.1854e-11 3.2054e-15 5.0745e-16
$ N^{-1}\eta $ 1.2815e-2 1.4300e-5 3.6844e-09 3.1940e-13 1.1690e-16
$ N^{-2}\eta $ 1.6018e-3 1.1917e-6 2.3027e-10 1.5970e-14 4.8708e-18
$ N $ 8 12 16 20 24
$ \|u-u_N\|_{L^2(\Omega)} $ 3.2745e-4 1.2301e-7 1.3536e-11 7.5344e-16 3.6242e-16
$ \|y-y_N\|_{L^2(\Omega)} $ 2.0412e-4 1.6475e-7 3.1550e-11 2.1122e-15 3.0444e-17
$ \|p-p_N\|_{L^2(\Omega)} $ 1.6373e-4 6.1497e-8 6.7677e-12 3.3969e-16 1.1459e-16
$ \|y-y_N\|_{H^1(\Omega)} $ 2.3942e-3 2.8825e-6 7.3202e-10 6.0843e-14 1.1835e-16
$ \|p-p_N\|_{H^1(\Omega)} $ 1.9464e-3 1.0822e-6 1.5764e-10 8.5485e-15 3.9737e-16
$ \Sigma_1 $ 4.6681e-3 4.0878e-6 9.0320e-10 7.0145e-14 8.7814e-16
$ \Sigma_2 $ 6.9530e-4 3.4925e-7 5.1854e-11 3.2054e-15 5.0745e-16
$ N^{-1}\eta $ 1.2815e-2 1.4300e-5 3.6844e-09 3.1940e-13 1.1690e-16
$ N^{-2}\eta $ 1.6018e-3 1.1917e-6 2.3027e-10 1.5970e-14 4.8708e-18
Table 2.  Error results between the numerical solutions and the exact solutions of Example 5.2
$ N $ 8 12 16 20 24
$ \|u-u_N\|_{L^2(\Omega)} $ 3.2510e-4 1.8937e-7 2.1653e-8 4.0625e-9 1.0698e-9
$ \|y-y_N\|_{L^2(\Omega)} $ 7.2724e-4 6.3163e-5 1.3710e-5 4.6053e-6 1.9404e-6
$ \|p-p_N\|_{L^2(\Omega)} $ 1.6255e-4 9.5404e-8 1.0913e-8 2.0474e-9 5.3917e-10
$ \Sigma_2 $ 1.2149e-3 6.3448e-5 1.3743e-5 4.6114e-6 1.9420e-6
$ N^{-2}\eta $ 2.9559e-3 1.8906e-4 3.8194e-5 1.2477e-5 5.1749e-6
$ N $ 8 12 16 20 24
$ \|u-u_N\|_{L^2(\Omega)} $ 3.2510e-4 1.8937e-7 2.1653e-8 4.0625e-9 1.0698e-9
$ \|y-y_N\|_{L^2(\Omega)} $ 7.2724e-4 6.3163e-5 1.3710e-5 4.6053e-6 1.9404e-6
$ \|p-p_N\|_{L^2(\Omega)} $ 1.6255e-4 9.5404e-8 1.0913e-8 2.0474e-9 5.3917e-10
$ \Sigma_2 $ 1.2149e-3 6.3448e-5 1.3743e-5 4.6114e-6 1.9420e-6
$ N^{-2}\eta $ 2.9559e-3 1.8906e-4 3.8194e-5 1.2477e-5 5.1749e-6
Table 3.  Error results between the numerical solutions and the exact solutions of Example 5.3
$ N $ 8 12 16 20 24
$ \|u-u_N\|_{L^2(\Omega)} $ 4.5350e-4 1.7023e-7 1.9481e-11 1.1412e-15 1.0292e-15
$ \|y-y_N\|_{L^2(\Omega)} $ 2.4022e-5 8.8248e-9 9.8738e-13 5.7858e-17 6.2448e-17
$ \|p-p_N\|_{L^2(\Omega)} $ 2.2675e-4 8.5110e-8 9.7411e-12 4.0934e-16 2.1952e-16
$ \|y-y_N\|_{H^1(\Omega)} $ 2.8912e-4 1.4819e-7 2.5124e-11 1.3216e-15 2.9444e-16
$ \|p-p_N\|_{H^1(\Omega)} $ 2.8451e-3 1.4558e-6 2.4692e-10 1.3310e-14 1.5921e-15
$ \Sigma_1 $ 3.5877e-3 1.7743e-6 2.9153e-10 1.5773e-14 2.9157e-15
$ \Sigma_2 $ 7.0427e-4 2.6416e-7 3.0210e-11 1.6084e-15 1.3112e-15
$ N^{-1}\eta $ 9.3412e-3 6.1876e-6 8.9996e-10 5.3721e-14 2.2490e-15
$ N^{-2}\eta $ 1.1676e-3 5.1563e-7 5.6247e-11 2.6861e-15 9.3707e-17
$ N $ 8 12 16 20 24
$ \|u-u_N\|_{L^2(\Omega)} $ 4.5350e-4 1.7023e-7 1.9481e-11 1.1412e-15 1.0292e-15
$ \|y-y_N\|_{L^2(\Omega)} $ 2.4022e-5 8.8248e-9 9.8738e-13 5.7858e-17 6.2448e-17
$ \|p-p_N\|_{L^2(\Omega)} $ 2.2675e-4 8.5110e-8 9.7411e-12 4.0934e-16 2.1952e-16
$ \|y-y_N\|_{H^1(\Omega)} $ 2.8912e-4 1.4819e-7 2.5124e-11 1.3216e-15 2.9444e-16
$ \|p-p_N\|_{H^1(\Omega)} $ 2.8451e-3 1.4558e-6 2.4692e-10 1.3310e-14 1.5921e-15
$ \Sigma_1 $ 3.5877e-3 1.7743e-6 2.9153e-10 1.5773e-14 2.9157e-15
$ \Sigma_2 $ 7.0427e-4 2.6416e-7 3.0210e-11 1.6084e-15 1.3112e-15
$ N^{-1}\eta $ 9.3412e-3 6.1876e-6 8.9996e-10 5.3721e-14 2.2490e-15
$ N^{-2}\eta $ 1.1676e-3 5.1563e-7 5.6247e-11 2.6861e-15 9.3707e-17
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