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 L^{2}(Ω) of the control variable is h | log h | in dimensions 2 and 3.

Keywords: optimal control, state constraints, elliptic equations, Borel measures, error estimates

@article{COCV_2014__20_3_803_0, author = {Casas, Eduardo and Mateos, Mariano and Vexler, Boris}, title = {New regularity results and improved error estimates for optimal control problems with state constraints}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {803--822}, publisher = {EDP-Sciences}, volume = {20}, number = {3}, year = {2014}, doi = {10.1051/cocv/2013084}, mrnumber = {3264224}, zbl = {1293.49044}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2013084/} }

TY - JOUR AU - Casas, Eduardo AU - Mateos, Mariano AU - Vexler, Boris TI - New regularity results and improved error estimates for optimal control problems with state constraints JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2014 SP - 803 EP - 822 VL - 20 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2013084/ DO - 10.1051/cocv/2013084 LA - en ID - COCV_2014__20_3_803_0 ER -

%0 Journal Article %A Casas, Eduardo %A Mateos, Mariano %A Vexler, Boris %T New regularity results and improved error estimates for optimal control problems with state constraints %J ESAIM: Control, Optimisation and Calculus of Variations %D 2014 %P 803-822 %V 20 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2013084/ %R 10.1051/cocv/2013084 %G en %F COCV_2014__20_3_803_0

Casas, Eduardo; Mateos, Mariano; Vexler, Boris. New regularity results and improved error estimates for optimal control problems with state constraints. ESAIM: Control, Optimisation and Calculus of Variations, Volume 20 (2014) no. 3, pp. 803-822. doi : 10.1051/cocv/2013084. http://archive.numdam.org/articles/10.1051/cocv/2013084/

[1] Error estimates for the numerical approximation of a semilinear elliptic control problem. Comput. Optim. Appl. 23 (2002) 201-229. | MR | Zbl

, and ,[2] Primal-dual strategy for state-constrained optimal control problems. Comput. Optim. Appl. 22 (2002) 193-224. | MR | Zbl

and ,[3] 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. | MR | Zbl

and ,[4] On the boundary value problem of the biharmonic operator on domains with angular corners. Math. Methods Appl. Sci. 2 (1980) 556-581. | MR | Zbl

and ,[5] The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York, Berlin, Heidelberg (1994). | MR | Zbl

and ,[6] Control of an elliptic problem with pointwise state constraints. SIAM J. Control Optim. 24 (1986) 1309-1318. | MR | Zbl

,[7] Sufficient second order optimality conditions for semilinear control problems with pointwise state constraints. SIAM J. Optim. 19 (2008) 616-643. | MR | Zbl

, and ,[8] Uniform convergence of the FEM. Applications to state constrained control problems. Comput. Appl. Math. 21 (2002) 67-100. | MR | Zbl

and ,[9] Error estimates for the numerical approximation of semilinear elliptic control problems with finitely many state contraints. ESAIM: COCV 8 (2002) 345-374. | Numdam | MR | Zbl

,[10] Numerical approximation of elliptic control problems with finitely many pointwise constraints. Comput. Optim. Appl. 51 (2012) 1319-1343. | MR | Zbl

and ,[11] Recent advances in the analysis of pointwise state-constrained elliptic optimal control problems. ESAIM: COCV 16 (2010) 581-600. | Numdam | MR | Zbl

and ,[12] Error estimates for the Lavrentiev regularization of elliptic optimal control problems. Inverse Problems 24 (2008) 21. | MR | Zbl

, and ,[13] Basic error estimates for elliptic problems, in Handbook of numerical analysis, Vol. II, Handb. Numer. Anal., II. North-Holland, Amsterdam (1991) 17-351 | MR | Zbl

,[14] Renormalized solutions of elliptic equations with general measure data. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 28 (1999) 741-808. | Numdam | MR | Zbl

, , and ,[15] Convergence of a finite element approximation to a state constrained elliptic control problem. SIAM J. Numer. Anal. 35 (2007) 1937-1953. | MR | Zbl

and ,[16] 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. | Zbl

and ,[17] A variational approach to semilinear elliptic equations with measure data. Discrete Contin. Dyn. Syst. 31 (2011) 1233-1248. | MR | Zbl

and ,[18] Elliptic partial differential equations of second order. Classics in Math. Reprint of the 1998 edition. Springer-Verlag, Berlin (2001). | MR | Zbl

and ,[19] A mixed finite element scheme for optimal control problems with pointwise state constraints. J. Sci. Comput. 46 (2011) 182-203. | MR | Zbl

and ,[20] Elliptic Problems in Nonsmooth Domains. Pitman, Boston-London-Melbourne, 1985. | MR | Zbl

,[21] Optimization with PDE constraints, vol. 23. Math. Model.: Theory Appl. Springer, New York (2009). | MR | Zbl

, , and ,[22] 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. | MR | Zbl

, and ,[23] A new finite element approximation of a state-constrained optimal control problem. J. Comput. Math. 27 (2009) 97-114. | MR | Zbl

, and ,[24] Error estimates for the finite-element approximation of an elliptic control problem with pointwise state and control constraints. Control Cybernet 37 (2008) 51-83. | MR | Zbl

,[25] C. Meyer, U. Prüfert and Tröltzsch, On two numerical methods for state-constrained elliptic control problems. Optim. Methods Softw. 22 (2007) 871-899. | MR | Zbl

[26] 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. | MR | Zbl

,[27] A priori error analysis for discretization of sparse elliptic optimal control problems in measure space. SIAM J. Control Optim. 51 (2013) 2788-2808. | MR

and ,[28] A priori error estimates for a state-constrained elliptic optimal control problem. ESAIM: M2AN 46 (2012) 1107-1120. | Numdam | Zbl

and ,[29] Real and Complex Analysis. McGraw-Hill, London (1970). | Zbl

,[30] Interior maximum norm estimates for finite element methods. Math. Comput. 31 (1977) 414-442. | MR | Zbl

and ,[31] Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier, Grenoble 15 (1965) 189-258. | Numdam | MR | Zbl

,*Cited by Sources: *