Error estimates for the numerical approximation of semilinear elliptic control problems with finitely many state constraints
ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002), pp. 345-374.

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

DOI : 10.1051/cocv:2002049
Classification : 49J20, 49K20, 49M05, 65K10
Mots-clés : distributed control, state constraints, semilinear elliptic equation, numerical approximation, finite element method, error estimates 
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Casas, Eduardo. Error estimates for the numerical approximation of semilinear elliptic control problems with finitely many state constraints. ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002), pp. 345-374. doi : 10.1051/cocv:2002049. https://www.numdam.org/articles/10.1051/cocv:2002049/

[1] N. Arada, E. Casas and F. Tröltzsch, Error estimates for the numerical approximation of a semilinear elliptic control problem. Comp. Optim. Appl. (to appear). | MR | Zbl

[2] V. Arnautu and P. Neittaanmäki, Discretization estimates for an elliptic control problem. Numer. Funct. Anal. Optim. (1998) 431-464. | MR | Zbl

[3] 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. | Zbl

[4] J. Bonnans and H. Zidani, Optimal control problems with partially polyhedric constraints. SIAM J. Control Optim. 37 (1999) 1726-1741. | MR | Zbl

[5] 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) 1431-1454. | MR | Zbl

[6] , Uniform convergence of the fem. applications to state constrained control problems. Comp. Appl. Math. 21 (2002). | MR | Zbl

[7] E. Casas, M. Mateos and L. Fernández, Second-order optimality conditions for semilinear elliptic control problems with constraints on the gradient of the state. Control Cybernet. 28 (1999) 463-479. | MR | Zbl

[8] E. Casas and F. Tröltzsch, Second order necessary optimality conditions for some state-constrained control problems of semilinear elliptic equations. App. Math. Optim. 39 (1999) 211-227. | MR | Zbl

[9] , Second order necessary and sufficient optimality conditions for optimization problems and applications to control theory. SIAM J. Optim. (to appear). | MR | Zbl

[10] E. Casas, F. Tröltzsch and A. Unger, Second order sufficient optimality conditions for a nonlinear elliptic control problem. J. Anal. Appl. 15 (1996) 687-707. | MR | Zbl

[11] , Second order sufficient optimality conditions for some state-constrained control problems of semilinear elliptic equations. SIAM J. Control Optim. 38 (2000) 1369-1391. | MR | Zbl

[12] P. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978). | MR | Zbl

[13] F. Clarke, A new approach to Lagrange multipliers. Math. Oper. Res. 1 (1976) 165-174. | MR | Zbl

[14] R. Falk, Approximation of a class of optimal control problems with order of convergence estimates. J. Math. Anal. Appl. 44 (1973) 28-47. | MR | Zbl

[15] T. Geveci, On the approximation of the solution of an optimal control problem governed by an elliptic equation. RAIRO: Numer. Anal. 13 (1979) 313-328. | Numdam | MR | Zbl

[16] H. Goldberg and F. Tröltzsch, Second order sufficient optimality conditions for a class of nonlinear parabolic boundary control problems. SIAM J. Control Optim. 31 (1993) 1007-1025. | MR | Zbl

[17] P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman, Boston-London-Melbourne (1985). | MR | Zbl

[18] 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. | MR | Zbl

[19] 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).

[20] P. Raviart and J. Thomas, Introduction à L'analyse Numérique des Equations aux Dérivées Partielles. Masson, Paris (1983). | Zbl

[21] J. Raymond and F. Tröltzsch, Second order sufficient optimality conditions for nonlinear parabolic control problems with state-constraints. Discrete Contin. Dynam. Systems 6 (2000) 431-450. | MR | Zbl

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