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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]

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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

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[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

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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

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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

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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

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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

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