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Error Estimates for the Numerical Approximationof SemilinearElliptic Control Problemswith Finitely Many State Constraints

Published online by Cambridge University Press:  15 August 2002

Eduardo Casas
Eduardo Casas*
Affiliation:
Dpt. Matemática Aplicada y Ciencias de la Computación, E.T.S.I.I y T., Universidad de Cantabria, Av. Los Castros s/n, 39005 Santander, Spain; [email protected].
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Abstract

The goal of this paper is to derive some error estimates for thenumerical discretization of some optimal control problems governedby semilinear elliptic equations with bound constraints on thecontrol and a finitely number of equality and inequality stateconstraints. We prove some error estimates for the optimalcontrols in the L ∞ norm and we also obtain error estimatesfor the Lagrange multipliers associated to the state constraintsas well as for the optimal states and optimal adjoint states.

Keywords

Distributed controlstate constraintssemilinear elliptic equationnumerical approximationfinite element methoderror estimates.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 8: A tribute to JL Lions , 2002 , pp. 345 - 374
Copyright
© EDP Sciences, SMAI, 2002

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References

N. Arada, E. Casas and F. Tröltzsch, Error estimates for the numerical approximation of a semilinear elliptic control problem. Comp. Optim. Appl. (to appear).
V. Arnautu and P. Neittaanmäki, Discretization estimates for an elliptic control problem. Numer. Funct. Anal. Optim. (1998) 431-464.
J. Bonnans and E. Casas, Contrôle de systèmes elliptiques semilinéaires comportant des contraintes sur l'état, in Nonlinear Partial Differential Equations and Their Applications, Vol. 8, Collège de France Seminar, edited by H. Brezis and J. Lions. Longman Scientific & Technical, New York (1988) 69-86.
Bonnans, J. and Zidani, H., Optimal control problems with partially polyhedric constraints. SIAM J. Control Optim. 37 (1999) 1726-1741. CrossRef
Casas, E. and Mateos, M., Second order optimality conditions for semilinear elliptic control problems with finitely many state constraints. SIAM J. Control Optim. 40 (2002) 1431-1454. CrossRef
height 2pt depth -1.6pt width 23pt, Uniform convergence of the fem. applications to state constrained control problems. Comp. Appl. Math. 21 (2002).
Casas, E., Mateos, M. and Fernández, L., Second-order optimality conditions for semilinear elliptic control problems with constraints on the gradient of the state. Control Cybernet. 28 (1999) 463-479.
Casas, E. and Tröltzsch, F., Second order necessary optimality conditions for some state-constrained control problems of semilinear elliptic equations. App. Math. Optim. 39 (1999) 211-227. CrossRef
height 2pt depth -1.6pt width 23pt, Second order necessary and sufficient optimality conditions for optimization problems and applications to control theory. SIAM J. Optim. (to appear).
Casas, E., Tröltzsch, F. and Unger, A., Second order sufficient optimality conditions for a nonlinear elliptic control problem. J. Anal. Appl. 15 (1996) 687-707.
height 2pt depth -1.6pt width 23pt, Second, order sufficient optimality conditions for some state-constrained control problems of semilinear elliptic equations. SIAM J. Control Optim. 38 (2000) 1369-1391.
P. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978).
Clarke, F., A new approach to Lagrange multipliers. Math. Oper. Res. 1 (1976) 165-174. CrossRef
Falk, R., Approximation of a class of optimal control problems with order of convergence estimates. J. Math. Anal. Appl. 44 (1973) 28-47. CrossRef
Geveci, T., On the approximation of the solution of an optimal control problem governed by an elliptic equation. RAIRO: Numer. Anal. 13 (1979) 313-328.
Goldberg, H. and Tröltzsch, F., Second order sufficient optimality conditions for a class of nonlinear parabolic boundary control problems. SIAM J. Control Optim. 31 (1993) 1007-1025. CrossRef
P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman, Boston-London-Melbourne (1985).
K. Malanowski, C. Büskens and H. Maurer, Convergence of approximations to nonlinear control problems, in Mathematical Programming with Data Perturbation, edited by A. Fiacco. New York, Marcel Dekker, Inc. (1997) 253-284.
M. Mateos, Problemas de control óptimo gobernados por ecuaciones semilineales con restricciones de tipo integral sobre el gradiente del estado, Ph.D. Thesis. University of Cantabria (2000).
P. Raviart and J. Thomas, Introduction à L'analyse Numérique des Equations aux Dérivées Partielles. Masson, Paris (1983).
Raymond, J. and Tröltzsch, F., Second order sufficient optimality conditions for nonlinear parabolic control problems with state-constraints. Discrete Contin. Dynam. Systems 6 (2000) 431-450.