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A unified theory of maximum principle for continuous and discrete time optimal control problems
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Approximate controllability of semilinear reaction diffusion equations
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 |
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. |
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. |
[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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