Error estimates for the numerical approximation of a semilinear elliptic control problem

Computational Optimization and Applications

Volumn 23, Issue 2, 2002, Pages 201-229

  Arada, Nadir ^a   Casas, Eduardo ^b   Tröltzsch, Fredi ^c  

^a CHEMNITZ UNIVERSITY OF TECHNOLOGY   (Germany)

^b UNIVERSITY OF CANTABRIA   (Spain)

^c TECHNISCHE UNIVERSITÄT BERLIN   (Germany)

Author keywords

Distributed control; Error estimates; Finite element method; Numerical approximation; Semilinear elliptic equation

Indexed keywords

APPROXIMATION THEORY; CONSTRAINT THEORY; CONVERGENCE OF NUMERICAL METHODS; ERROR ANALYSIS; FINITE ELEMENT METHOD; INTEGRAL EQUATIONS; MATHEMATICAL OPERATORS; OPTIMAL CONTROL SYSTEMS; PARTIAL DIFFERENTIAL EQUATIONS; THEOREM PROVING;

DISTRIBUTED NONLINEAR CONTROL PROBLEM; ERROR ESTIMATE; SEMILINEAR ELLIPTIC EQUATION;

OPTIMIZATION;

EID: 0036853826     PISSN: 09266003     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1020576801966     Document Type: Article

References (32)

1. W. Alt and U. Mackenroth, "Convergence of finite element approximations to state constrained convex parabolic boundary control problems," SIAM J. Control and Optim., vol. 27, pp. 718-736, 1989.
2. N. Arada and J.-P. Raymond, "Approximation of Young measures for state-constrained optimal control problems," Numer. Funct. Anal. and Optimiz., vol. 21, 2000.
3. V. Arnautu and P. Neittaanmäki, "Discretization estimates for an elliptic control problem," Numer. Funct. Anal. and Optimiz., vol. 19, pp. 431-464, 1998.
4. J.F. Bonnans and E. Casas, "An extension of Pontryagin's principle for state constrained optimal control of semilinear elliptic and variational inequalities," SIAM J. Control and Optim., vol. 33, pp. 274-298, 1995.
5. S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, Springer, 1991.
6. E. Casas, "Optimal control in coefficients of elliptic equations with state constraints," Appl. Math. Optim., vol. 26, pp. 21-37, 1992.
7. E. Casas, "Finite element approximations for some optimal control problems with state constraints," Pn IMACS 91, 13th World Congress on Computation and Applied Math., R. Vichnevetski and J. Miller (Eds.), Dublin, 3, pp. 1165-1166, 1991.
8. E. Casas and M. Mateos, "Second order optimality conditions for semilinear elliptic control problems with finitely many state constraints," SIAM J. Control and Optim., vol. 40, pp. 1431-1454, 2002.
9. E. Casas and M. Mateos, "Uniform convergence of the FEM. Applications to state constrained control problems," Comp. Appl. Math., to appear.
10. E. Casas and F. Tröltzsch, "Second order necessary and sufficient optimality conditions for optimization problems and applications to control theory," SIAM J. Optim., to appear.
11. P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North Holland, 1978.
12. A.L. Dontchev and W.W. Hager, "The Euler approximation in state constrained optimal control," Math. of Computation, vol. 70, pp. 173-203, 2000.
13. A.L. Dontchev and W.W. Hager, "Second-order Runge-Kutta approximations in control constrained optimal control," SIAM J. Numer. Anal., vol. 38, pp. 202-226, 2000.
14. R. Falk, "Approximation of a class of optimal control problems with order of convergence estimates," J. Math. Anal. Appl., vol. 44, pp. 28-47, 1973.
15. T. Geveci, "On the approximation of the solution of an optimal control problem problem governed by an elliptic equation," R.A.I.R.O. Analyse numérique/Numerical Analysis., vol. 13, pp. 313-328, 1979.
16. P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, 1985.
17. W.W. Hager, "Multiplier methods for nonlinear optimal control," SIAM J. Numer. Anal., vol. 27, pp. 1061-1080, 1990.
18. W.W. Hager, "Numerical analysis in optimal control," in: Optimal Control of Complex Structures. International Series of Numerical Mathematics, Birkhäuser, Basel, vol. 139, pp. 83-93, 2001.
19. G. Knowles, "Finite element approximation of parabolic time optimal control problems," SIAM J. Control and Optim., vol. 20, pp. 414-427, 1982.
20. I. Lasiecka, "Boundary control of parabolic systems: Finite-element approximations," Appl. Math. Optim., vol. 6 pp. 287-333, 1980.
21. I. Lasiecka, "Ritz-Galerkin approximation of the time optimal boundary control problem for parabolic systems with Dirichlet boundary conditions," SIAM J. Control and Optim., vol. 97, pp. 477-500, 1984.
22. U. Mackenroth, "Some remarks on the numerical solution of bang-bang-type optimal control problems," Numer. Funct. Anal. and Optimiz., vol. 5, pp. 457-484, 1982-1983.
23. U. Mackenroth, "Numerical solutions of some parabolic boundary control problems by finite elements," in Control Problems for Systems Described by Partial Differential Equations and Applications. Lect. Notes Contr. Inf. Sci., vol. 97, pp. 325-335, 1987.
24. 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, pp. 253-284, 1997.
25. M. Mateos, Problemas de control óptimo gobernados por ecuaciones semilineales con restricciones de tipo integral sobre el gradiente del estado, Thesis, University of Cantabria, Santander, June 2000.
26. R.S. McKnight and W.E. Bosarge, "The Ritz-Galerkin procedure for parabolic control problems," Siam J. Control, vol. 11, pp. 510-524, 1973.
27. P.A. Raviart and J.M. Thomas, "Introduction à l'Analyse Numérique des Équations aux Dérivées Partielles, Masson, 1992.
28. D. Tiba and F. Tröltzsch, "Error estimates for the discretization of state constrained convex control problems," Numer. Funct. Anal. and Optimiz., vol. 17, pp. 1005-1028, 1996.
29. F. Tröltzsch, "Semidiscrete finite element approximation of parabolic boundary control problems-convergence of switching points," in: Optimal Control of Partial Differential Equations II. International Series of Numerical Mathematics, Birkhäuser, Basel, vol. 78, pp. 219-232, 1987.
30. F. Tröltzsch, "Approximation of nonlinear parabolic boundary problems by the Fourier method-Convergence of optimal controls," Optimization, vol. 22, pp. 83-98, 1991.
31. F. Tröltzsch, "On a convergence of semidiscrete Ritz-Galerkin schemes applied to the boundary control of parabolic equations with non-linear boundary condition," Z. Angew. Math. Mech, vol. 72, pp. 291-301, 1992.
32. F. Tröltzsch, "Semidiscrete Ritz-Galerkin approximation of nonlinear parabolic boundary control problems-Strong convergence of optimal controls," Appl. Math. Optim., vol. 29, pp. 309-329, 1994.