Error estimates for the numerical approximation of dirichlet boundary control for semilinear elliptic equations

SIAM Journal on Control and Optimization

Volumn 45, Issue 5, 2006, Pages 1586-1611

  Casas, Eduardo ^a   Raymond, Jean Pierre ^b  

^a UNIVERSITY OF CANTABRIA   (Spain)

^b INSTITUT DE MATHÉMATIQUES DE TOULOUSE   (France)

Author keywords

Dirichlet control; Error estimates; Numerical approximation; Semilinear elliptic equation

Indexed keywords

APPROXIMATION ALGORITHMS; BOUNDARY CONDITIONS; COMPUTATIONAL GEOMETRY; COMPUTER SIMULATION; ERROR ANALYSIS; FINITE ELEMENT METHOD; STATE ESTIMATION;

DIRICHLET CONTROL; PIECEWISE LINEAR FINITE ELEMENTS; SEMILINEAR ELLIPTIC EQUATIONS;

PARTIAL DIFFERENTIAL EQUATIONS;

EID: 34547960540     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/050626600     Document Type: Article

References (41)

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), pp. 201-229.
2. N. ARADA AND J.-P. RAYMOND, Dirichlet boundary control of semilinear parabolic equations. Part 1: Problems with no state constraints, Appl. Math. Optim., 45 (2002), pp. 125-143.
3. M. BERGGREN, Approximations of very weak solutions to boundary-value problems, SIAM J. Numer. Anal., 42 (2004), pp. 860-877.
4. J. F. BONNANS AND H. ZIDANI, Optimal control problems with partially polyhedric constraints, SIAM J. Control Optim., 37 (1999), pp. 1726-1741.
5. J. BRAMBLE, J. PASCIAK, AND A. SCHATZ, The construction of preconditioned for elliptic problems by substructuring. I, Math. Comp., 47 (1986), pp. 103-134.
6. S. BRENNER AND L. R. SCOTT, The Mathematical Theory of Finite Element Methods, Springer-Verlag, New York, Berlin, Heidelberg, 1994.
7. H. BRÉZIS, Analyse Fonctionnelle, Théorie et Applications, Masson, Paris, 1983.
8. E. CASAS, Error estimates for the numerical approximation of semilinear elliptic control problems with finitely many state constraints, ESAIM Control Optim. Calc. Var., 8 (2002), pp. 345-374.
9. E. CASAS AND M. MATEOS, Second order optimality conditions for semilinear elliptic control problems with finitely many state constraints, SIAM J. Control Optim., 40 (2002), pp. 1431-1454.
10. E. CASAS AND M. MATEOS, Uniform convergence of the FEM. Applications to state constrained control problems, Comput. Appl. Math., 21 (2002), pp. 67-100.
11. E. CASAS, M. MATEOS, AND F. TRÖLTZSCH, Error estimates for the numerical approximation of boundary semilinear elliptic control problems, Comput. Optim. Appl., 31 (2005), pp. 193-219.
12. E. CASAS, M. MATEOS, AND J.-P. RAYMOND, Robin approximation of Dirichlet optimal control problems for semilinear elliptic equations, in preparation.
13. 2-projections on some convex sets, Numer. Funct. Anal. Optim., 27 (2006), pp. 117-137.
14. P. CIARLET AND P. RAVIART, Maximum principle and uniform convergence for the finite element method, Comput. Methods Appl. Mech. Engrg., 2 (1973), pp. 17-31.
15. M. DAUGE, Neumann and mixed problems on curvilinear polyhedra, Integral Equations Operator Theory, 15 (1992), 227-261.
16. A. DONTCHEV AND W. HAGER, The Euler approximation in state constrained optimal control, Math. Comp., 70 (2000), pp. 173-203.
17. A. L. DONTCHEV, W. HAGER, AND V. M. VELIOV, Second-order Runge-Kutta approximations in control constrained optimal control, SIAM J. Numer. Anal., 38 (2000), pp. 202-226.
18. 2 projection into finite element function spaces, Numer. Math., 23 (1975), pp. 193-197.
19. R. FALK, Approximation of a class of optimal control problems with order of convergence estimates, J. Math. Anal. Appl., 44 (1973), pp. 28-47.
20. D. FRENCH AND J. KING, Approximation of an elliptic control problem by the finite element method, Numer. Funct, Anal. Optim., 12 (1991), pp. 299-314.
21. T. GEVECI, On the approximation of the solution of an optimal control problem governed by an elliptic equation, RAIRO Anal. Numér., 13 (1979), pp. 313-328.
22. P. GRISVARD, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, London, Melbourne, 1985.
23. M. D. GUNZBURGER, L. S. HOU, AND TH. P. SVOBODNY, Analysis and finite element approximation of optimal control problems for the stationary Navier-Stokes equations with Dirichlet controls, RAIRO Modél. Math. Anal. Numér., 25 (1991), pp. 711-748.
24. W. W. HAGER, Multiplier methods for nonlinear optimal control, SIAM J. Numer. Anal., 27 (1990), pp. 1061-1080.
25. W. HAGER, Numerical analysis in optimal control, in Optimal Control of Complex Structures, Internat. Ser. Numer. Math. 139, Birkhäuser, Basel, 2002, pp. 83-93.
26. M. HINZE, A variational discretization concept in control constrained optimization: The linear-quadratic case, Comput. Optim. Appl., 30 (2005), pp. 45-61.
27. G. KNOWLES, Finite element approximation of parabolic time optimal control problems, SIAM J. Control Optim., 20 (1982), pp. 414-427.
28. I. LASIECKA, Boundary control of parabolic systems: Finite-element approximations, Appl. Math. Optim., 6 (1980), pp. 287-333.
29. I. LASIECKA, Ritz-Galerkin approximation of the time optimal boundary control problem for parabolic systems with Dirichlet boundary conditions, SIAM J. Control Optim., 22 (1984), pp. 477-500.
30. J. L. LIONS AND E. MAGENES, Problèmes aux Limites non Homogènes, Dunod, Paris, 1968.
31. K. MALANOWSKI, C. BÜSKENS, AND H. MAURER, Convergence of approximations to nonlinear control problems, in Mathematical Programming with Data Perturbation, A. V. Fiacco, ed., Marcel Dekker, Inc., New York, 1997, pp. 253-284.
32. R. S. MCKNIGHT AND W. E. BOSARGE, Jr., The Ritz-Galerkin procedure for parabolic control problems, SIAM J. Control Optim., 11 (1973), pp. 510-524.
33. C. MEYER AND A. RÖSCH, Superconvergence properties of optimal control problems, SIAM J. Control Optim., 43 (2004), pp. 970-985.
34. G. STAMPACCHIA, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), pp. 189-258.
35. L. R. SCOTT AND S. ZHANG, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp., 54 (1990), pp. 483-493.
36. D. TIBA AND F. TRÖLTZSCH, Error estimates for the discretization of state constrained convex control problems, Numer. Funct. Anal. Optim., 17 (1996), pp. 1005-1028.
37. H. TRIEBEL, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978.
38. F. TRÖLTZSCH, Semidiscrete finite element approximation of parabolic boundary control problems - convergence of switching points, in Optimal Control of Partial Differential Equations II, Internat. Ser. Numer. Math. 78, Birkhäuser, Basel, 1987, pp. 219-232.
39. F. TRÖLTZSCH, Approximation of nonlinear parabolic boundary problems by the Fourier method - convergence of optimal controls, Optimization, 2 (1991), pp. 83-98.
40. F. TRÖLTZSCH, On convergence of semidiscrete Ritz-Galerkin schemes applied to the boundary control of parabolic equations with nonlinear boundary condition, Z. Angew. Math. Mech., 72 (1992), pp. 291-301.
41. F. TRÖLTZSCH, Semidiscrete Ritz-Galerkin approximation of nonlinear parabolic boundary control problems - strong convergence of optimal controls, Appl. Math. Optim., 29 (1994), pp. 309-329.