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Error analysis for global minima of semilinear optimal control problems
1. | Schwerpunkt Optimierung und Approximation, Universität Hamburg, Bundesstraße 55,20146 Hamburg, Germany |
2. | Institut für Analysis und Numerik, Otto-von-Guericke-Universität Magdeburg, Universitätsplatz 2,39106 Magdeburg, Germany |
In [
References:
[1] |
A. Ahmad Ali, Optimal Control of Semilinear Elliptic PDEs with State Constraints -Numerical Analysis and Implementation, PhD thesis, Dissertation, Hamburg, Universität Hamburg, 2017. |
[2] |
A. Ahmad Ali, K. Deckelnick and M. Hinze,
Global minima for semilinear optimal control problems, Computational Optimization and Applications, 65 (2016), 261-288.
|
[3] |
N. Arada, E. Casas and F. Tröltzsch,
Error estimates for the numerical approximation of a semilinear elliptic control problem, Computational Optimization and Applications, 23 (2002), 201-229.
|
[4] |
E. Casas,
L2 Estimates for the finite element method for the Dirichlet problem with singular data, Numerische Mathematik, 47 (1985), 627-632.
|
[5] |
E. Casas,
Control of an elliptic problem with pointwise state constraints, SIAM Journal on Control and Optimization, 24 (1986), 1309-1318.
|
[6] |
E. Casas,
Boundary control of semilinear elliptic equations with pointwise state constraints, SIAM Journal on Control and Optimization, 31 (1993), 993-1006.
|
[7] |
E. Casas,
Error estimates for the numerical approximation of semilinear elliptic control problems with finitely many state constraints, ESAIM: Control, Optimisation and Calculus of Variations, 8 (2002), 345-374.
|
[8] |
E. Casas,
Necessary and sufficient optimality conditions for elliptic control problems with finitely many pointwise state constraints, ESAIM: Control, Optimisation and Calculus of Variations, 14 (2008), 575-589.
|
[9] |
E. Casas, J. C. De Los Reyes and F. Tröltzsch,
Sufficient second-order optimality conditions for semilinear control problems with pointwise state constraints, SIAM Journal on Optimization, 19 (2008), 616-643.
|
[10] |
E. Casas and L. A. Fernández,
Optimal control of semilinear elliptic equations with pointwise constraints on the gradient of the state, Applied Mathematics and Optimization, 27 (1993), 35-56.
|
[11] |
E. Casas and M. Mateos,
Second order optimality conditions for semilinear elliptic control problems with finitely many state constraints, SIAM Journal on Control and Optimization, 40 (2002), 1431-1454.
|
[12] |
E. Casas and M. Mateos, Uniform convergence of the FEM. Applications to state constrained control problems, Comput. Appl. Math., 21 (2002), 67-100, Special issue in memory of Jacques-Louis Lions. |
[13] |
E. Casas, M. Mateos and B. Vexler,
New regularity results and improved error estimates for optimal control problems with state constraints, ESAIM. Control, Optimisation and Calculus of Variations, 20 (2014), 803-822.
|
[14] |
E. Casas and F. Tröltzsch,
Recent advances in the analysis of pointwise state-constrained elliptic optimal control problems, ESAIM: Control, Optimisation and Calculus of Variations, 16 (2010), 581-600.
|
[15] |
E. Casas and F. Tröltzsch,
Second order optimality conditions and their role in pde control, Jahresbericht der Deutschen Mathematiker-Vereinigung, 117 (2015), 3-44.
|
[16] |
K. Deckelnick and M. Hinze,
Convergence of a finite element approximation to a state-constrained elliptic control problem, SIAM Journal on Numerical Analysis, 45 (2007), 1937-1953.
|
[17] |
K. Deckelnick and M. Hinze, A finite element approximation to elliptic control problems in the presence of control and state constraints, Hamburger Beiträge zur Angewandten Mathematik, (2007). |
[18] |
P. Grisvard, Elliptic Problems in Nonsmooth Domains, vol. 69, SIAM, 2011. |
[19] |
M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints, vol. 23 of Mathematical Modelling: Theory and Applications, Springer, New York, 2009. |
[20] |
M. Hinze,
A variational discretization concept in control constrained optimization: The linear-quadratic case, Computational Optimization and Applications, 30 (2005), 45-61.
|
[21] |
M. Hinze and C. Meyer,
Stability of semilinear elliptic optimal control problems with pointwise state constraints, Computational Optimization and Applications, 52 (2012), 87-114.
|
[22] |
M. Hinze and A. Rösch, Discretization of optimal control problems, in Constrained Optimization and Optimal Control for Partial Differential Equations, Springer, 160 (2012), 391-430. |
[23] |
M. Hinze and F. Tröltzsch,
Discrete concepts versus error analysis in pde-constrained optimization, GAMM-Mitteilungen, 33 (2010), 148-162.
|
[24] |
D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Classics in Applied Mathematics, 31. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. |
[25] |
P. Merino, F. Tröltzsch and B. Vexler,
Error estimates for the finite element approximation of a semilinear elliptic control problem with state constraints and finite dimensional control space, ESAIM: Mathematical Modelling and Numerical Analysis, 44 (2010), 167-188.
|
[26] |
C. Meyer,
Error estimates for the finite-element approximation of an elliptic control problem with pointwise state and control constraints, Control and Cybernetics, 37 (2008), 51-83.
|
[27] |
I. Neitzel, J. Pfefferer and A. Rösch,
Finite element discretization of state-constrained elliptic optimal control problems with semilinear state equation, SIAM Journal on Control and Optimization, 53 (2015), 874-904.
|
[28] |
I. Neitzel and W. Wollner, A priori L2-discretization error estimates for the state in elliptic optimization problems with pointwise inequality state constraints, Numer. Math., online first (2017). |
show all references
References:
[1] |
A. Ahmad Ali, Optimal Control of Semilinear Elliptic PDEs with State Constraints -Numerical Analysis and Implementation, PhD thesis, Dissertation, Hamburg, Universität Hamburg, 2017. |
[2] |
A. Ahmad Ali, K. Deckelnick and M. Hinze,
Global minima for semilinear optimal control problems, Computational Optimization and Applications, 65 (2016), 261-288.
|
[3] |
N. Arada, E. Casas and F. Tröltzsch,
Error estimates for the numerical approximation of a semilinear elliptic control problem, Computational Optimization and Applications, 23 (2002), 201-229.
|
[4] |
E. Casas,
L2 Estimates for the finite element method for the Dirichlet problem with singular data, Numerische Mathematik, 47 (1985), 627-632.
|
[5] |
E. Casas,
Control of an elliptic problem with pointwise state constraints, SIAM Journal on Control and Optimization, 24 (1986), 1309-1318.
|
[6] |
E. Casas,
Boundary control of semilinear elliptic equations with pointwise state constraints, SIAM Journal on Control and Optimization, 31 (1993), 993-1006.
|
[7] |
E. Casas,
Error estimates for the numerical approximation of semilinear elliptic control problems with finitely many state constraints, ESAIM: Control, Optimisation and Calculus of Variations, 8 (2002), 345-374.
|
[8] |
E. Casas,
Necessary and sufficient optimality conditions for elliptic control problems with finitely many pointwise state constraints, ESAIM: Control, Optimisation and Calculus of Variations, 14 (2008), 575-589.
|
[9] |
E. Casas, J. C. De Los Reyes and F. Tröltzsch,
Sufficient second-order optimality conditions for semilinear control problems with pointwise state constraints, SIAM Journal on Optimization, 19 (2008), 616-643.
|
[10] |
E. Casas and L. A. Fernández,
Optimal control of semilinear elliptic equations with pointwise constraints on the gradient of the state, Applied Mathematics and Optimization, 27 (1993), 35-56.
|
[11] |
E. Casas and M. Mateos,
Second order optimality conditions for semilinear elliptic control problems with finitely many state constraints, SIAM Journal on Control and Optimization, 40 (2002), 1431-1454.
|
[12] |
E. Casas and M. Mateos, Uniform convergence of the FEM. Applications to state constrained control problems, Comput. Appl. Math., 21 (2002), 67-100, Special issue in memory of Jacques-Louis Lions. |
[13] |
E. Casas, M. Mateos and B. Vexler,
New regularity results and improved error estimates for optimal control problems with state constraints, ESAIM. Control, Optimisation and Calculus of Variations, 20 (2014), 803-822.
|
[14] |
E. Casas and F. Tröltzsch,
Recent advances in the analysis of pointwise state-constrained elliptic optimal control problems, ESAIM: Control, Optimisation and Calculus of Variations, 16 (2010), 581-600.
|
[15] |
E. Casas and F. Tröltzsch,
Second order optimality conditions and their role in pde control, Jahresbericht der Deutschen Mathematiker-Vereinigung, 117 (2015), 3-44.
|
[16] |
K. Deckelnick and M. Hinze,
Convergence of a finite element approximation to a state-constrained elliptic control problem, SIAM Journal on Numerical Analysis, 45 (2007), 1937-1953.
|
[17] |
K. Deckelnick and M. Hinze, A finite element approximation to elliptic control problems in the presence of control and state constraints, Hamburger Beiträge zur Angewandten Mathematik, (2007). |
[18] |
P. Grisvard, Elliptic Problems in Nonsmooth Domains, vol. 69, SIAM, 2011. |
[19] |
M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints, vol. 23 of Mathematical Modelling: Theory and Applications, Springer, New York, 2009. |
[20] |
M. Hinze,
A variational discretization concept in control constrained optimization: The linear-quadratic case, Computational Optimization and Applications, 30 (2005), 45-61.
|
[21] |
M. Hinze and C. Meyer,
Stability of semilinear elliptic optimal control problems with pointwise state constraints, Computational Optimization and Applications, 52 (2012), 87-114.
|
[22] |
M. Hinze and A. Rösch, Discretization of optimal control problems, in Constrained Optimization and Optimal Control for Partial Differential Equations, Springer, 160 (2012), 391-430. |
[23] |
M. Hinze and F. Tröltzsch,
Discrete concepts versus error analysis in pde-constrained optimization, GAMM-Mitteilungen, 33 (2010), 148-162.
|
[24] |
D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Classics in Applied Mathematics, 31. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. |
[25] |
P. Merino, F. Tröltzsch and B. Vexler,
Error estimates for the finite element approximation of a semilinear elliptic control problem with state constraints and finite dimensional control space, ESAIM: Mathematical Modelling and Numerical Analysis, 44 (2010), 167-188.
|
[26] |
C. Meyer,
Error estimates for the finite-element approximation of an elliptic control problem with pointwise state and control constraints, Control and Cybernetics, 37 (2008), 51-83.
|
[27] |
I. Neitzel, J. Pfefferer and A. Rösch,
Finite element discretization of state-constrained elliptic optimal control problems with semilinear state equation, SIAM Journal on Control and Optimization, 53 (2015), 874-904.
|
[28] |
I. Neitzel and W. Wollner, A priori L2-discretization error estimates for the state in elliptic optimization problems with pointwise inequality state constraints, Numer. Math., online first (2017). |




Levels | EOC |
EOC |
EOC |
EOC |
2-3 | 1.187645 | 0.833842 | 1.464788 | 1.058334 |
3-4 | 1.078183 | 0.948273 | 1.708822 | 1.758387 |
4-5 | 1.027290 | 0.985352 | 1.794456 | 1.657899 |
5-6 | 1.016702 | 0.997996 | 1.831198 | 1.514376 |
Levels | EOC |
EOC |
EOC |
EOC |
2-3 | 1.187645 | 0.833842 | 1.464788 | 1.058334 |
3-4 | 1.078183 | 0.948273 | 1.708822 | 1.758387 |
4-5 | 1.027290 | 0.985352 | 1.794456 | 1.657899 |
5-6 | 1.016702 | 0.997996 | 1.831198 | 1.514376 |
Levels | EOC |
EOC |
EOC |
EOC |
2-3 | 1.910131 | 1.015420 | 1.686197 | 1.399920 |
3-4 | 1.983722 | 1.017850 | 1.934978 | 1.876141 |
4-5 | 1.982064 | 1.005320 | 1.996849 | 1.993434 |
5-6 | 1.944839 | 1.003374 | 2.035042 | 2.027269 |
Levels | EOC |
EOC |
EOC |
EOC |
2-3 | 1.910131 | 1.015420 | 1.686197 | 1.399920 |
3-4 | 1.983722 | 1.017850 | 1.934978 | 1.876141 |
4-5 | 1.982064 | 1.005320 | 1.996849 | 1.993434 |
5-6 | 1.944839 | 1.003374 | 2.035042 | 2.027269 |
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