# Error estimates for the finite element discretization of semi-infinite elliptic optimal control problems

Pedro Merino; Ira Neitzel; Fredi Tröltzsch

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2010)

- Volume: 30, Issue: 2, page 221-236
- ISSN: 1509-9407

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topPedro Merino, Ira Neitzel, and Fredi Tröltzsch. "Error estimates for the finite element discretization of semi-infinite elliptic optimal control problems." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 30.2 (2010): 221-236. <http://eudml.org/doc/271164>.

@article{PedroMerino2010,

abstract = {In this paper we derive a priori error estimates for linear-quadratic elliptic optimal control problems with finite dimensional control space and state constraints in the whole domain, which can be written as semi-infinite optimization problems. Numerical experiments are conducted to ilustrate our theory.},

author = {Pedro Merino, Ira Neitzel, Fredi Tröltzsch},

journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},

keywords = {elliptic optimal control problem; state constraints; error estimates; finite element discretization},

language = {eng},

number = {2},

pages = {221-236},

title = {Error estimates for the finite element discretization of semi-infinite elliptic optimal control problems},

url = {http://eudml.org/doc/271164},

volume = {30},

year = {2010},

}

TY - JOUR

AU - Pedro Merino

AU - Ira Neitzel

AU - Fredi Tröltzsch

TI - Error estimates for the finite element discretization of semi-infinite elliptic optimal control problems

JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization

PY - 2010

VL - 30

IS - 2

SP - 221

EP - 236

AB - In this paper we derive a priori error estimates for linear-quadratic elliptic optimal control problems with finite dimensional control space and state constraints in the whole domain, which can be written as semi-infinite optimization problems. Numerical experiments are conducted to ilustrate our theory.

LA - eng

KW - elliptic optimal control problem; state constraints; error estimates; finite element discretization

UR - http://eudml.org/doc/271164

ER -

## References

top- [1] N. Arada, E. Casas and F. Tröltzsch, Error estimates for the numerical approximation of a semilinear elliptic control problem, Comput. Optim. Appl. 23 (2002), 201-229. doi: 10.1023/A:1020576801966 Zbl1033.65044
- [2] F. Bonnans and A. Shapiro, Perturbation analysis of optimization problems (Springer, New York, 2000). Zbl0966.49001
- [3] E. Casas, Using piecewise linear functions in the numerical approximation of semilinear elliptic control problems, Advances in Computational Mathematics 26 (2007), 137-153. doi: 10.1007/s10444-004-4142-0 Zbl1118.65069
- [4] E. Casas and M. Mateos, Error estimates for the numerical approximation of Neumann control problems, Comput. Optim. Appl. 39 (2008), 265-295. doi: 10.1007/s10589-007-9056-6 Zbl1191.49030
- [5] P.G. Ciarlet, The Finite Element Method for Elliptic Problems (North-Holland, Amsterdam, 1978). Zbl0383.65058
- [6] K. Deckelnick and M. Hinze, Convergence of a finite element approximation to a state constrained elliptic control problem, SIAM J. Numer. Anal. 45 (2007), 1937-1953. Zbl1154.65055
- [7] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order (Springer, 3rd edition, 1998). Zbl1042.35002
- [8] P. Grisvard, Elliptic Problems in Nonsmooth Domains (Pitman, Boston, 1985). Zbl0695.35060
- [9] G. Gramlich, R. Hettich and E.W. Sachs, Local convergence of SQP methods in semi-infinite programming, SIAM J. Optim. 5 (1995), 641-658. Zbl0840.65056
- [10] M. Hinze, A variational discretization concept in control constrained optimization: the linear-quadratic case, J. Comput. Optim. Appl. 30 (2005), 45-63. doi: 10.1137/060652361 Zbl1074.65069
- [11] M. Huth and R. Tichatschke, A hybrid method for semi-infinite programming problems, Operations research, Proc. 14th Symp. Ulm/FRG 1989, Methods Oper. Res. 62 (1990), 79-90.
- [12] 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:M2AN 44 (1) (2010), 167-188. Zbl1191.65076
- [13] C. Meyer, Error estimates for the finite-element approximation of an elliptic control problem with pointwise state and control constraints, Control Cybern. 37 (2008), 51-85. Zbl1170.65055
- [14] C. Meyer and A. Rösch, Superconvergence properties of optimal control problems, SIAM J. Control and Optimization 43 (2004), 970-985. Zbl1071.49023
- [15] C. Meyer, U. Prüfert and F. Tröltzsch, On two numerical methods for state-constrained elliptic control problems, Optimization Methods and Software 22 (6) (2007), 871-899. Zbl1172.49022
- [16] R. Rannacher and B. Vexler, A priori error estimates for the finite element discretization of elliptic parameter identification problems with pointwise measurements, SIAM Control Optim. 44 (2005), 1844-1863. Zbl1113.65102
- [17] R. Reemtsen and J.-J. Rückmann (Eds), Semi-Infinite Programming (Kluwer Academic Publishers, Boston, 1998). doi: 10.1007/978-1-4757-2868-2
- [18] A. R{ösch, Error estimates for linear-quadratic control problems with control constraints, Optimization Methods and Software 21 (1) (2006), 121-134. doi: 10.1080/10556780500094945 Zbl1085.49042
- [19] G. Still, Discretization in semi-infinite programming: the rate of convergence, Mathematical Programming. A Publication of the Mathematical Programming Society 91 (1) (A) (2001), 53-69. Zbl1051.90023
- [20] G. Still, Generalized semi-infinite programming: Numerical aspects, Optimization 49 (3) (2001), 223-242. Zbl1039.90083
- [21] F. Guerra Vázquez, J.-J. Rückmann, O. Stein and G. Still, Generalized semi-infinite programming: a tutorial, J. Comput. Appl. Math. 217 (2008), 394-419. doi: 10.1016/j.cam.2007.02.012 Zbl1190.90248

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