Error estimates for the finite element discretization of semi-infinite elliptic optimal control problems
Discussiones Mathematicae. Differential Inclusions, Control and Optimization, Tome 30 (2010) no. 2, pp. 221-236.

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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.
Keywords: elliptic optimal control problem, state constraints, error estimates, finite element discretization
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Merino, Pedro; Neitzel, Ira; Tröltzsch, Fredi. Error estimates for the finite element discretization of semi-infinite elliptic optimal control problems. Discussiones Mathematicae. Differential Inclusions, Control and Optimization, Tome 30 (2010) no. 2, pp. 221-236. http://geodesic.mathdoc.fr/item/DMDICO_2010_30_2_a3/

[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

[2] F. Bonnans and A. Shapiro, Perturbation analysis of optimization problems (Springer, New York, 2000).

[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

[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

[5] P.G. Ciarlet, The Finite Element Method for Elliptic Problems (North-Holland, Amsterdam, 1978).

[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.

[7] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order (Springer, 3rd edition, 1998).

[8] P. Grisvard, Elliptic Problems in Nonsmooth Domains (Pitman, Boston, 1985).

[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.

[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

[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.

[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.

[14] C. Meyer and A. Rösch, Superconvergence properties of optimal control problems, SIAM J. Control and Optimization 43 (2004), 970-985.

[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.

[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.

[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

[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.

[20] G. Still, Generalized semi-infinite programming: Numerical aspects, Optimization 49 (3) (2001), 223-242.

[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