# American Institute of Mathematical Sciences

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

 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.
Citation: Ming Yan, Lili Chang, Ningning Yan. Finite element method for constrained optimal control problems governed by nonlinear elliptic PDEs. Mathematical Control and 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. [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. [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. [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. [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. [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. [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. [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. [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. [10] M. Hintermüller, A proximal bundle method based on approximate subgradients, Computational Optimization and Applications, 20 (2001), 245-266. doi: 10.1023/A:1011259017643. [11] M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, "Optimization with PDE Constraints," Mathematical Modelling: Theory and Applications, 23, Springer, New York, 2009. [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. [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. [14] R. Li and W. B. Liu, AFEPack, Numerical software. Available from: http://dsec.pku.edu.cn/~rli/software_e.php. [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. [16] W. B. Liu and N. Yan, "Adaptive Finite Element Methods for Optimal Control Governed by PDEs," Science Press, Beijing, 2008. [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. [18] D. Tiba, "Lectures on the Optimal Control of Elliptic Equations," University of Jyvaskyla Press, Finland, 1995.

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##### 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. [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. [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. [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. [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. [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. [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. [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. [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. [10] M. Hintermüller, A proximal bundle method based on approximate subgradients, Computational Optimization and Applications, 20 (2001), 245-266. doi: 10.1023/A:1011259017643. [11] M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, "Optimization with PDE Constraints," Mathematical Modelling: Theory and Applications, 23, Springer, New York, 2009. [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. [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. [14] R. Li and W. B. Liu, AFEPack, Numerical software. Available from: http://dsec.pku.edu.cn/~rli/software_e.php. [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. [16] W. B. Liu and N. Yan, "Adaptive Finite Element Methods for Optimal Control Governed by PDEs," Science Press, Beijing, 2008. [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. [18] D. Tiba, "Lectures on the Optimal Control of Elliptic Equations," University of Jyvaskyla Press, Finland, 1995.
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