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2010 | 30 | 2 | 221-236
Tytuł artykułu

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

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EN
Abstrakty
EN
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.
Twórcy
autor
  • Technische Universität Berlin, Institut für Mathematik, Germany
  • Escuela Politécnica Nacional, Departamento de Matemática, Ecuador
autor
  • Technische Universität Berlin, Institut für Mathematik, Germany
  • Technische Universität Berlin, Institut für Mathematik, Germany
Bibliografia
  • [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
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1121
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