American Institute of Mathematical Sciences

Advanced
June  2012, 2(2): 183-194. doi: 10.3934/mcrf.2012.2.183

Finite element method for constrained optimal control problems governed by nonlinear elliptic PDEs

Ming Yan 1, , Lili Chang 1, and  Ningning Yan 1,

1. 

Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China, China, China

Received  March 2011 Revised  November 2011 Published  May 2012

In this paper, we study the finite element method for constrained optimal control problems governed by nonlinear elliptic PDEs. Instead of the standard error estimates under $L^2$- or $H^1$- norm, we apply the goal-oriented error estimates in order to avoid the difficulties which are generated by the nonsmoothness of the problem. We derive the a priori error estimates of the goal function, and the error bound is $O(h^2)$, which is the same as one for some well known quadratic optimal control problems governed by linear elliptic PDEs. Moreover, two kinds of practical algorithms are introduced to solve the underlying problem. Numerical experiments are provided to confirm our theoretical results.
Keywords: nonlinear elliptic PDEs, goal-oriented error estimates., finite element method, Optimal control problems.
Mathematics Subject Classification: Primary: 49J20, 65N15, 65N3.
Citation: Ming Yan, Lili Chang, Ningning Yan. Finite element method for constrained optimal control problems governed by nonlinear elliptic PDEs. Mathematical Control & Related Fields, 2012, 2 (2) : 183-194. doi: 10.3934/mcrf.2012.2.183
References:
[1]

A. K. Aziz, A. B. Stephens and M. Suri, Numerical methods for reaction-diffusion problems with nondifferentiable kinetics, Numer. Math., 53 (1988), 1-11. doi: 10.1007/BF01395875.  Google Scholar

[2]

R. Becker and R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods, Acta Numerica, 10 (2001), 1-102. doi: 10.1017/S0962492901000010.  Google Scholar

[3]

B. Bergounioux, K. Ito and K. Kunisch, Primal-dual strategy for constrained optimal control problems, SIAM J. Control Optim., 37 (1999), 1176-1194. doi: 10.1137/S0363012997328609.  Google Scholar

[4]

J. Burke, A. Lewis and M. Overton, A robust gradient sampling algorithm for nonsmooth, nonconvex optimization, SIAM J. Optim., 15 (2005), 751-779. doi: 10.1137/030601296.  Google Scholar

[5]

L. Chang, W. Gong and N. Yan, Finite element method for a nonsmooth elliptic equation, Frontiers of Mathematics in China, 5 (2010), 191-209.  Google Scholar

[6]

X. Chen, First order conditions for nonsmooth discretized constrained optimal control problems, SIAM J. Control Optim., 42 (2004), 2004-2015. doi: 10.1137/S0363012902414160.  Google Scholar

[7]

X. Chen, Z. Nashed and L. Qi, Smoothing methods and semismooth methods for nondifferentiable operator equations, SIAM J. Numer. Anal., 38 (2000), 1200-1216. doi: 10.1137/S0036142999356719.  Google Scholar

[8]

F. Clarke, "Optimization and Nonsmooth Analysis," Canadian Mathematical Society Series of Monographs and Advanced Texts, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983.  Google Scholar

[9]

F. S. Falk, Approximation of a class of optimal control problems with order of convergence estimates, J. Math. Anal. Appl., 44 (1973), 28-47. doi: 10.1016/0022-247X(73)90022-X.  Google Scholar

[10]

M. Hintermüller, A proximal bundle method based on approximate subgradients, Computational Optimization and Applications, 20 (2001), 245-266. doi: 10.1023/A:1011259017643.  Google Scholar

[11]

M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, "Optimization with PDE Constraints," Mathematical Modelling: Theory and Applications, 23, Springer, New York, 2009.  Google Scholar

[12]

F. Kikuchi, Finite element analysis of a nondifferentiable nonlinear problem related to MHD equilibria, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 35 (1988), 77-101.  Google Scholar

[13]

F. Kikuchi, K. Nakazato and T. Ushijima, Finite element approximation of a nonlinear eigenvalue problem related to MHD equilibria, Japan J. Appl. Math., 1 (1984), 369-403.  Google Scholar

[14]

R. Li and W. B. Liu, AFEPack, Numerical software., Available from: \url{http://dsec.pku.edu.cn/~rli/software_e.php}., ().   Google Scholar

[15]

J.-L. Lions, "Optimal Control of Systems Governed by Partial Differential Equations," Translated from the French by S. K. Mitter, Die Grundlehren der mathematischen Wissenschaften, Band 170, Springer-Verlag, New York-Berlin, 1971.  Google Scholar

[16]

W. B. Liu and N. Yan, "Adaptive Finite Element Methods for Optimal Control Governed by PDEs," Science Press, Beijing, 2008. Google Scholar

[17]

J. Shen, Z.-Q. Xia and L.-P. Pang, A proximal bundle method with inexact data for convex nondifferentiable minimization, Nonlinear Analysis, 66 (2007), 2016-2027. doi: 10.1016/j.na.2006.02.039.  Google Scholar

[18]

D. Tiba, "Lectures on the Optimal Control of Elliptic Equations," University of Jyvaskyla Press, Finland, 1995. Google Scholar

show all references

References:
[1]

A. K. Aziz, A. B. Stephens and M. Suri, Numerical methods for reaction-diffusion problems with nondifferentiable kinetics, Numer. Math., 53 (1988), 1-11. doi: 10.1007/BF01395875.  Google Scholar

[2]

R. Becker and R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods, Acta Numerica, 10 (2001), 1-102. doi: 10.1017/S0962492901000010.  Google Scholar

[3]

B. Bergounioux, K. Ito and K. Kunisch, Primal-dual strategy for constrained optimal control problems, SIAM J. Control Optim., 37 (1999), 1176-1194. doi: 10.1137/S0363012997328609.  Google Scholar

[4]

J. Burke, A. Lewis and M. Overton, A robust gradient sampling algorithm for nonsmooth, nonconvex optimization, SIAM J. Optim., 15 (2005), 751-779. doi: 10.1137/030601296.  Google Scholar

[5]

L. Chang, W. Gong and N. Yan, Finite element method for a nonsmooth elliptic equation, Frontiers of Mathematics in China, 5 (2010), 191-209.  Google Scholar

[6]

X. Chen, First order conditions for nonsmooth discretized constrained optimal control problems, SIAM J. Control Optim., 42 (2004), 2004-2015. doi: 10.1137/S0363012902414160.  Google Scholar

[7]

X. Chen, Z. Nashed and L. Qi, Smoothing methods and semismooth methods for nondifferentiable operator equations, SIAM J. Numer. Anal., 38 (2000), 1200-1216. doi: 10.1137/S0036142999356719.  Google Scholar

[8]

F. Clarke, "Optimization and Nonsmooth Analysis," Canadian Mathematical Society Series of Monographs and Advanced Texts, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983.  Google Scholar

[9]

F. S. Falk, Approximation of a class of optimal control problems with order of convergence estimates, J. Math. Anal. Appl., 44 (1973), 28-47. doi: 10.1016/0022-247X(73)90022-X.  Google Scholar

[10]

M. Hintermüller, A proximal bundle method based on approximate subgradients, Computational Optimization and Applications, 20 (2001), 245-266. doi: 10.1023/A:1011259017643.  Google Scholar

[11]

M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, "Optimization with PDE Constraints," Mathematical Modelling: Theory and Applications, 23, Springer, New York, 2009.  Google Scholar

[12]

F. Kikuchi, Finite element analysis of a nondifferentiable nonlinear problem related to MHD equilibria, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 35 (1988), 77-101.  Google Scholar

[13]

F. Kikuchi, K. Nakazato and T. Ushijima, Finite element approximation of a nonlinear eigenvalue problem related to MHD equilibria, Japan J. Appl. Math., 1 (1984), 369-403.  Google Scholar

[14]

R. Li and W. B. Liu, AFEPack, Numerical software., Available from: \url{http://dsec.pku.edu.cn/~rli/software_e.php}., ().   Google Scholar

[15]

J.-L. Lions, "Optimal Control of Systems Governed by Partial Differential Equations," Translated from the French by S. K. Mitter, Die Grundlehren der mathematischen Wissenschaften, Band 170, Springer-Verlag, New York-Berlin, 1971.  Google Scholar

[16]

W. B. Liu and N. Yan, "Adaptive Finite Element Methods for Optimal Control Governed by PDEs," Science Press, Beijing, 2008. Google Scholar

[17]

J. Shen, Z.-Q. Xia and L.-P. Pang, A proximal bundle method with inexact data for convex nondifferentiable minimization, Nonlinear Analysis, 66 (2007), 2016-2027. doi: 10.1016/j.na.2006.02.039.  Google Scholar

[18]

D. Tiba, "Lectures on the Optimal Control of Elliptic Equations," University of Jyvaskyla Press, Finland, 1995. Google Scholar

[1]

Dominik Hafemeyer, Florian Mannel, Ira Neitzel, Boris Vexler. Finite element error estimates for one-dimensional elliptic optimal control by BV-functions. Mathematical Control & Related Fields, 2020, 10 (2) : 333-363. doi: 10.3934/mcrf.2019041
[2]

Lijuan Wang, Jun Zou. Error estimates of finite element methods for parameter identifications in elliptic and parabolic systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1641-1670. doi: 10.3934/dcdsb.2010.14.1641
[3]

Marita Holtmannspötter, Arnd Rösch, Boris Vexler. A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system. Mathematical Control & Related Fields, 2021, 11 (3) : 601-624. doi: 10.3934/mcrf.2021014
[4]

Philip Trautmann, Boris Vexler, Alexander Zlotnik. Finite element error analysis for measure-valued optimal control problems governed by a 1D wave equation with variable coefficients. Mathematical Control & Related Fields, 2018, 8 (2) : 411-449. doi: 10.3934/mcrf.2018017
[5]

Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077
[6]

Liupeng Wang, Yunqing Huang. Error estimates for second-order SAV finite element method to phase field crystal model. Electronic Research Archive, 2021, 29 (1) : 1735-1752. doi: 10.3934/era.2020089
[7]

Chunjuan Hou, Yanping Chen, Zuliang Lu. Superconvergence property of finite element methods for parabolic optimal control problems. Journal of Industrial & Management Optimization, 2011, 7 (4) : 927-945. doi: 10.3934/jimo.2011.7.927
[8]

Tao Lin, Yanping Lin, Weiwei Sun. Error estimation of a class of quadratic immersed finite element methods for elliptic interface problems. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 807-823. doi: 10.3934/dcdsb.2007.7.807
[9]

Jie Shen, Xiaofeng Yang. Error estimates for finite element approximations of consistent splitting schemes for incompressible flows. Discrete & Continuous Dynamical Systems - B, 2007, 8 (3) : 663-676. doi: 10.3934/dcdsb.2007.8.663
[10]

Assyr Abdulle, Yun Bai, Gilles Vilmart. Reduced basis finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems. Discrete & Continuous Dynamical Systems - S, 2015, 8 (1) : 91-118. doi: 10.3934/dcdss.2015.8.91
[11]

Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2581-2598. doi: 10.3934/dcdsb.2020196
[12]

Eduardo Casas, Mariano Mateos, Arnd Rösch. Finite element approximation of sparse parabolic control problems. Mathematical Control & Related Fields, 2017, 7 (3) : 393-417. doi: 10.3934/mcrf.2017014
[13]

Martin Burger, José A. Carrillo, Marie-Therese Wolfram. A mixed finite element method for nonlinear diffusion equations. Kinetic & Related Models, 2010, 3 (1) : 59-83. doi: 10.3934/krm.2010.3.59
[14]

Hee-Dae Kwon, Jeehyun Lee, Sung-Dae Yang. Eigenseries solutions to optimal control problem and controllability problems on hyperbolic PDEs. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 305-325. doi: 10.3934/dcdsb.2010.13.305
[15]

Donald L. Brown, Vasilena Taralova. A multiscale finite element method for Neumann problems in porous microstructures. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1299-1326. doi: 10.3934/dcdss.2016052
[16]

Qingping Deng. A nonoverlapping domain decomposition method for nonconforming finite element problems. Communications on Pure & Applied Analysis, 2003, 2 (3) : 297-310. doi: 10.3934/cpaa.2003.2.297
[17]

Runchang Lin. A robust finite element method for singularly perturbed convection-diffusion problems. Conference Publications, 2009, 2009 (Special) : 496-505. doi: 10.3934/proc.2009.2009.496
[18]

Xiu Ye, Shangyou Zhang, Peng Zhu. A weak Galerkin finite element method for nonlinear conservation laws. Electronic Research Archive, 2021, 29 (1) : 1897-1923. doi: 10.3934/era.2020097
[19]

Tianliang Hou, Yanping Chen. Superconvergence for elliptic optimal control problems discretized by RT1 mixed finite elements and linear discontinuous elements. Journal of Industrial & Management Optimization, 2013, 9 (3) : 631-642. doi: 10.3934/jimo.2013.9.631
[20]

Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. II. Convergence of the method of finite differences. Inverse Problems & Imaging, 2016, 10 (4) : 869-898. doi: 10.3934/ipi.2016025

2019 Impact Factor: 0.857

Tools

Metrics

  • PDF downloads (28)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences