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.  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.  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.  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.  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.   Google Scholar

[6]

X. Chen, First order conditions for nonsmooth discretized constrained optimal control problems,, SIAM J. Control Optim., 42 (2004), 2004.  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.  doi: 10.1137/S0036142999356719.  Google Scholar

[8]

F. Clarke, "Optimization and Nonsmooth Analysis,", Canadian Mathematical Society Series of Monographs and Advanced Texts, (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.  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.  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 (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.   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.   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, (1971).   Google Scholar

[16]

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

[6]

X. Chen, First order conditions for nonsmooth discretized constrained optimal control problems,, SIAM J. Control Optim., 42 (2004), 2004.  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.  doi: 10.1137/S0036142999356719.  Google Scholar

[8]

F. Clarke, "Optimization and Nonsmooth Analysis,", Canadian Mathematical Society Series of Monographs and Advanced Texts, (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.  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.  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 (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.   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.   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, (1971).   Google Scholar

[16]

W. B. Liu and N. Yan, "Adaptive Finite Element Methods for Optimal Control Governed by PDEs,", Science Press, (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.  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, (1995).   Google Scholar

[1]

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
[2]

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
[3]

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
[4]

Hongbo Guan, Yong Yang, Huiqing Zhu. A nonuniform anisotropic FEM for elliptic boundary layer optimal control problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1711-1722. doi: 10.3934/dcdsb.2020179
[5]

Wenya Qi, Padmanabhan Seshaiyer, Junping Wang. A four-field mixed finite element method for Biot's consolidation problems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020127
[6]

Ying Liu, Yanping Chen, Yunqing Huang, Yang Wang. Two-grid method for semiconductor device problem by mixed finite element method and characteristics finite element method. Electronic Research Archive, 2021, 29 (1) : 1859-1880. doi: 10.3934/era.2020095
[7]

Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019
[8]

Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang. A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Research Archive, , () : -. doi: 10.3934/era.2020120
[9]

Bin Wang, Lin Mu. Viscosity robust weak Galerkin finite element methods for Stokes problems. Electronic Research Archive, 2021, 29 (1) : 1881-1895. doi: 10.3934/era.2020096
[10]

Christian Clason, Vu Huu Nhu, Arnd Rösch. Optimal control of a non-smooth quasilinear elliptic equation. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020052
[11]

Ole Løseth Elvetun, Bjørn Fredrik Nielsen. A regularization operator for source identification for elliptic PDEs. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021006
[12]

Gang Bao, Mingming Zhang, Bin Hu, Peijun Li. An adaptive finite element DtN method for the three-dimensional acoustic scattering problem. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 61-79. doi: 10.3934/dcdsb.2020351
[13]

Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046
[14]

Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107
[15]

Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Fractional optimal control problems on a star graph: Optimality system and numerical solution. Mathematical Control & Related Fields, 2021, 11 (1) : 189-209. doi: 10.3934/mcrf.2020033
[16]

Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168
[17]

Liping Tang, Ying Gao. Some properties of nonconvex oriented distance function and applications to vector optimization problems. Journal of Industrial & Management Optimization, 2021, 17 (1) : 485-500. doi: 10.3934/jimo.2020117
[18]

Waixiang Cao, Lueling Jia, Zhimin Zhang. A $ C^1 $ Petrov-Galerkin method and Gauss collocation method for 1D general elliptic problems and superconvergence. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 81-105. doi: 10.3934/dcdsb.2020327
[19]

Elimhan N. Mahmudov. Infimal convolution and duality in convex optimal control problems with second order evolution differential inclusions. Evolution Equations & Control Theory, 2021, 10 (1) : 37-59. doi: 10.3934/eect.2020051
[20]

Lars Grüne, Roberto Guglielmi. On the relation between turnpike properties and dissipativity for continuous time linear quadratic optimal control problems. Mathematical Control & Related Fields, 2021, 11 (1) : 169-188. doi: 10.3934/mcrf.2020032

2019 Impact Factor: 0.857

Tools

Metrics

  • PDF downloads (23)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences