Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-10T16:45:17.675Z Has data issue: false hasContentIssue false
  • English
  • Français

New regularity results and improved error estimates for optimal control problems with state constraints∗

Published online by Cambridge University Press:  05 June 2014

Eduardo Casas ,
Mariano Mateos  and
Boris Vexler
Eduardo Casas
Affiliation:
Departmento de Matemática Aplicada y Ciencias de la Computación, E.T.S.I. Industriales y de Telecomunicación, Universidad de Cantabria, 39005 Santander, Spain. eduardo.casas@unican.es
Mariano Mateos
Affiliation:
Departmento de Matemáticas, E.P.I. Gijón, Universidad de Oviedo, Campus de Gijón, 33203 Gijón, Spain; mmateos@uniovi.es
Boris Vexler
Affiliation:
Center for Mathematical Sciences, Technische Universität München, Bolzmannstrasse 3, 85748 Garching b. München, Germany; vexler@ma.tum.de
Get access

Abstract

In this paper we are concerned with a distributed optimal control problem governed by an elliptic partial differential equation. State constraints of box type are considered. We show that the Lagrange multiplier associated with the state constraints, which is known to be a measure, is indeed more regular under quite general assumptions. We discretize the problem by continuous piecewise linear finite elements and we are able to prove that, for the case of a linear equation, the order of convergence for the error in L2(Ω) of the control variable is h | log h | in dimensions 2 and 3.

Keywords

Optimal controlstate constraintselliptic equationsBorel measureserror estimates
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 3 , July 2014 , pp. 803 - 822
Copyright
© EDP Sciences, SMAI 2014

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

Arada, N., Casas, E. and Tröltzsch, F., Error estimates for the numerical approximation of a semilinear elliptic control problem. Comput. Optim. Appl. 23 (2002) 201–229. Google Scholar
Bergounioux, M. and Kunisch, K., Primal-dual strategy for state-constrained optimal control problems. Comput. Optim. Appl. 22 (2002) 193–224. Google Scholar
Bergounioux, M. and Kunisch, K., On the structure of Lagrange multipliers for state-constrained optimal control problems. Systems Control Lett. 48 (2003) 169–176. Optimization and control of distributed systems. Google Scholar
Blum, H. and Rannacher, R., On the boundary value problem of the biharmonic operator on domains with angular corners. Math. Methods Appl. Sci. 2 (1980) 556–581. Google Scholar
S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York, Berlin, Heidelberg (1994).
Casas, E., Control of an elliptic problem with pointwise state constraints. SIAM J. Control Optim. 24 (1986) 1309–1318. Google Scholar
Casas, E., de los Reyes, J.C. and Tröltzsch, F., Sufficient second order optimality conditions for semilinear control problems with pointwise state constraints. SIAM J. Optim. 19 (2008) 616–643. Google Scholar
Casas, E. and Mateos, M., Uniform convergence of the FEM. Applications to state constrained control problems. Comput. Appl. Math. 21 (2002) 67–100. Google Scholar
Casas, E., Error estimates for the numerical approximation of semilinear elliptic control problems with finitely many state contraints. ESAIM: COCV 8 (2002) 345–374. Google Scholar
Casas, E. and Mateos, M., Numerical approximation of elliptic control problems with finitely many pointwise constraints. Comput. Optim. Appl. 51 (2012) 1319–1343. Google Scholar
Casas, E. and Tröltzsch, F., Recent advances in the analysis of pointwise state-constrained elliptic optimal control problems. ESAIM: COCV 16 (2010) 581–600. Google Scholar
Cherednichenko, S., Krumbiegel, K. and Rösch, A., Error estimates for the Lavrentiev regularization of elliptic optimal control problems. Inverse Problems 24 (2008) 21. Google Scholar
P.G. Ciarlet, Basic error estimates for elliptic problems, in Handbook of numerical analysis, Vol. II, Handb. Numer. Anal., II. North-Holland, Amsterdam (1991) 17–351
Dal Maso, G., Murat, F., Orsina, L. and Prignet, A., Renormalized solutions of elliptic equations with general measure data. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 28 (1999) 741–808. Google Scholar
Deckelnick, K. and Hinze, M., Convergence of a finite element approximation to a state constrained elliptic control problem. SIAM J. Numer. Anal. 35 (2007) 1937–1953. Google Scholar
K. Deckelnick and M. Hinze, Numerical analysis of a control and state constrained elliptic control problem with piecewise constant control approximations, in Proc. of ENUMATH, 2007. Numer. Math. Advanced Appl., edited by K. Kunisch, G. Of and O. Steinbach. Springer, Berlin (2008) 597–604.
Degiovanni, M. and Scaglia, M., A variational approach to semilinear elliptic equations with measure data. Discrete Contin. Dyn. Syst. 31 (2011) 1233–1248. Google Scholar
D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order. Classics in Math. Reprint of the 1998 edition. Springer-Verlag, Berlin (2001).
Gong, W. and Yan, N., A mixed finite element scheme for optimal control problems with pointwise state constraints. J. Sci. Comput. 46 (2011) 182–203. Google Scholar
P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman, Boston-London-Melbourne, 1985.
M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE constraints, vol. 23. Math. Model.: Theory Appl. Springer, New York (2009).
Leykekhman, D., Meidner, D. and Vexler, B., Optimal error estimates for finite element discretization of elliptic optimal control problems with finitely many pointwise state constraints. Comput. Optim. Appl. 55 (2013) 769–802. Google Scholar
Liu, W., Gong, W. and Yan, N., A new finite element approximation of a state-constrained optimal control problem. J. Comput. Math. 27 (2009) 97–114. Google Scholar
Meyer, C., Error estimates for the finite-element approximation of an elliptic control problem with pointwise state and control constraints. Control Cybernet 37 (2008) 51–83. Google Scholar
Meyer, C., U. Prüfert and Tröltzsch, On two numerical methods for state-constrained elliptic control problems. Optim. Methods Softw. 22 (2007) 871–899. Google Scholar
Meyer, C., Error estimates for the finite-element approximation of an elliptic control problem with pointwise state and control constraints. Control and Cybernetics 37 (2008) 51–85. Google Scholar
Pieper, K. and Vexler, B., A priori error analysis for discretization of sparse elliptic optimal control problems in measure space. SIAM J. Control Optim. 51 (2013) 2788–2808. Google Scholar
Rösch, A. and Steinig, S., A priori error estimates for a state-constrained elliptic optimal control problem. ESAIM: M2AN 46 (2012) 1107–1120. Google Scholar
W. Rudin, Real and Complex Analysis. McGraw-Hill, London (1970).
Schatz, A.H. and Wahlbin, L.B., Interior maximum norm estimates for finite element methods. Math. Comput. 31 (1977) 414–442. Google Scholar
Stampacchia, G., Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier, Grenoble 15 (1965) 189–258. Google Scholar