Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-08T05:30:40.330Z Has data issue: false hasContentIssue false
  • English
  • Français

Recent advances in the analysis of pointwise state-constrainedelliptic optimal control problems

Published online by Cambridge University Press:  02 July 2009

Eduardo Casas  and
Fredi Tröltzsch
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]
Fredi Tröltzsch
Affiliation:
Institut für Mathematik, Technische Universität Berlin, 10623 Berlin, Germany. [email protected]
Get access

Abstract

Optimal control problems for semilinear elliptic equationswith control constraints and pointwise state constraints arestudied. Several theoretical results are derived, which arenecessary to carry out a numerical analysis for this class ofcontrol problems. In particular, sufficient second-order optimalityconditions, some new regularity results on optimal controls and asufficient condition for the uniqueness of the Lagrange multiplierassociated with the state constraints are presented.

Keywords

Optimal controlpointwise state constraintsfirst and second order optimality conditionsLagrange multipliersBorel measures
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 16 , Issue 3 , July 2010 , pp. 581 - 600
Copyright
© EDP Sciences, SMAI, 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alibert, J.-J. and Raymond, J.-P., Boundary control of semilinear elliptic equations with discontinuous leading coefficients and unbounded controls. Numer. Funct. Anal. Optim. 3-4 (1997) 235–250. CrossRef
Casas, E., Boundary control of semilinear elliptic equations with pointwise state constraints. SIAM J. Control Optim. 31 (1993) 993–1006. CrossRef
Casas, E., Error estimates for the numerical approximation of semilinear elliptic control problems with finitely many state constraints. ESAIM: COCV 8 (2002) 345–374. CrossRef
Casas, E., Necessary and sufficient optimality conditions for elliptic control problems with finitely many pointwise state constraints. ESAIM: COCV 14 (2008) 575–589. 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
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
E. Casas, J.-P. Raymond and H. Zidani, 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.
Casas, E., Tröltzsch, F. and Unger, A., Second order sufficient optimality conditions for some state-constrained control problems of semilinear elliptic equations. SIAM J. Control Optim. 38 (2000) 1369–1391. CrossRef
Casas, E., de los Reyes, J. and Tröltzsch, F., Sufficient second order optimality conditions for semilinear control problems with pointwise state constraints. SIAM J. Optim. 19 (2008) 616–643. CrossRef
Deckelnick, M. and Hinze, M., Convergence of a finite element approximation to a state-constrained elliptic control problem. SIAM J. Numer. Anal. 45 (2007) 1937–1953. CrossRef
M. Deckelnick and M. Hinze, 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.
D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin-Heidelberg-New York (1977).
P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman, Boston-London-Melbourne (1985).
Jerison, D. and Kenig, C., The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal. 130 (1995) 161–219. CrossRef
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, Spain (2000).
Maurer, H. and Zowe, J., First- and second-order conditions in infinite-dimensional programming problems. Math. Programming 16 (1979) 98–110. CrossRef
P. Merino, F. Tröltzsch and B. Vexler, Error estimates for the finite element approximation of a semilinear elliptic control problem with state constraints and finite dimensional control space. ESAIM: M2AN (submitted).
Meyer, C., Error estimates for the finite-element approximation of an elliptic control problem with pointwise state and control constraints. Control Cybern. 37 (2008) 51–83.
Saut, J. and Scheurer, B., 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. CrossRef
Stampacchia, G., Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble) 15 (1965) 189–258. CrossRef