Optimal control problems for semilinear elliptic equations with control constraints and pointwise state constraints are studied. Several theoretical results are derived, which are necessary to carry out a numerical analysis for this class of control problems. In particular, sufficient second-order optimality conditions, some new regularity results on optimal controls and a sufficient condition for the uniqueness of the Lagrange multiplier associated with the state constraints are presented.
Mots clés : optimal control, pointwise state constraints, first and second order optimality conditions, Lagrange multipliers, Borel measures
@article{COCV_2010__16_3_581_0,
author = {Casas, Eduardo and Tr\"oltzsch, Fredi},
title = {Recent advances in the analysis of pointwise state-constrained elliptic optimal control problems},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {581--600},
publisher = {EDP-Sciences},
volume = {16},
number = {3},
year = {2010},
doi = {10.1051/cocv/2009010},
mrnumber = {2674627},
zbl = {1201.49004},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/cocv/2009010/}
}
TY - JOUR AU - Casas, Eduardo AU - Tröltzsch, Fredi TI - Recent advances in the analysis of pointwise state-constrained elliptic optimal control problems JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2010 SP - 581 EP - 600 VL - 16 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2009010/ DO - 10.1051/cocv/2009010 LA - en ID - COCV_2010__16_3_581_0 ER -
%0 Journal Article %A Casas, Eduardo %A Tröltzsch, Fredi %T Recent advances in the analysis of pointwise state-constrained elliptic optimal control problems %J ESAIM: Control, Optimisation and Calculus of Variations %D 2010 %P 581-600 %V 16 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2009010/ %R 10.1051/cocv/2009010 %G en %F COCV_2010__16_3_581_0
Casas, Eduardo; Tröltzsch, Fredi. Recent advances in the analysis of pointwise state-constrained elliptic optimal control problems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 16 (2010) no. 3, pp. 581-600. doi : 10.1051/cocv/2009010. http://archive.numdam.org/articles/10.1051/cocv/2009010/
[1] and , Boundary control of semilinear elliptic equations with discontinuous leading coefficients and unbounded controls. Numer. Funct. Anal. Optim. 3-4 (1997) 235-250. | MR | Zbl
[2] , Boundary control of semilinear elliptic equations with pointwise state constraints. SIAM J. Control Optim. 31 (1993) 993-1006. | MR | Zbl
[3] , Error estimates for the numerical approximation of semilinear elliptic control problems with finitely many state constraints. ESAIM: COCV 8 (2002) 345-374. | EuDML | Numdam | MR | Zbl
[4] , Necessary and sufficient optimality conditions for elliptic control problems with finitely many pointwise state constraints. ESAIM: COCV 14 (2008) 575-589. | EuDML | Numdam | MR
[5] and , 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] and , Second order necessary optimality conditions for some state-constrained control problems of semilinear elliptic equations. App. Math. Optim. 39 (1999) 211-227. | MR | Zbl
[7] , and , Optimal control problems governed by semilinear elliptic equations with integral control constraints and pointwise state constraints, in International Conference on Control and Estimations of Distributed Parameter Systems, Vorau, Austria, 1996, W. Desch, F. Kappel and K. Kunisch Eds., Int. Series Num. Analysis, Birkhäuser-Verlag, Basel (1998) 89-102. | MR | Zbl
[8] , and , 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
[9] , and , Sufficient second order optimality conditions for semilinear control problems with pointwise state constraints. SIAM J. Optim. 19 (2008) 616-643. | MR | Zbl
[10] and , Convergence of a finite element approximation to a state-constrained elliptic control problem. SIAM J. Numer. Anal. 45 (2007) 1937-1953. | Zbl
[11] and , Numerical analysis of a control and state constrained elliptic control problem with piecewise constant control approximations, in Numerical Mathematics and Advanced Applications, Proceedings of ENUMATH 2007, Graz, Austria, K. Kunisch, G. Of and O. Steinbach Eds., Springer, Heidelberg (2008) 597-604. | Zbl
[12] and , Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin-Heidelberg-New York (1977). | Zbl
[13] , Elliptic Problems in Nonsmooth Domains. Pitman, Boston-London-Melbourne (1985). | Zbl
[14] and , The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal. 130 (1995) 161-219. | Zbl
[15] , Problemas de control óptimo gobernados por ecuaciones semilineales con restricciones de tipo integral sobre el gradiente del estado. Ph.D. Thesis, University of Cantabria, Spain (2000).
[16] and , First- and second-order conditions in infinite-dimensional programming problems. Math. Programming 16 (1979) 98-110. | Zbl
[17] , and , Error estimates for the finite element approximation of a semilinear elliptic control problem with state constraints and finite dimensional control space. ESAIM: M2AN (submitted). | Numdam | Zbl
[18] , Error estimates for the finite-element approximation of an elliptic control problem with pointwise state and control constraints. Control Cybern. 37 (2008) 51-83. | Zbl
[19] and , Sur l'unicité du problème de Cauchy et le prolongement unique pour des équations elliptiques à coefficients non localement bornés. J. Diff. Eq. 43 (1982) 28-43. | Zbl
[20] , Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble) 15 (1965) 189-258. | Numdam | Zbl
Cité par Sources :
