# A priori error estimates for a state-constrained elliptic optimal control problem

ESAIM: Mathematical Modelling and Numerical Analysis (2012)

- Volume: 46, Issue: 5, page 1107-1120
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topRösch, Arnd, and Steinig, Simeon. "A priori error estimates for a state-constrained elliptic optimal control problem." ESAIM: Mathematical Modelling and Numerical Analysis 46.5 (2012): 1107-1120. <http://eudml.org/doc/277841>.

@article{Rösch2012,

abstract = {We examine an elliptic optimal control problem with control and state constraints in
ℝ3. An improved error estimate of
𝒪(hs)
with 3/4 ≤ s ≤ 1 − ε is proven for a discretisation
involving piecewise constant functions for the control and piecewise linear for the state.
The derived order of convergence is illustrated by a numerical example.},

author = {Rösch, Arnd, Steinig, Simeon},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Elliptic optimal control problem; state constraint; a priori error estimates; elliptic control problem; state constrained; a priori error estimates; convergence rate; finite elements},

language = {eng},

month = {2},

number = {5},

pages = {1107-1120},

publisher = {EDP Sciences},

title = {A priori error estimates for a state-constrained elliptic optimal control problem},

url = {http://eudml.org/doc/277841},

volume = {46},

year = {2012},

}

TY - JOUR

AU - Rösch, Arnd

AU - Steinig, Simeon

TI - A priori error estimates for a state-constrained elliptic optimal control problem

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2012/2//

PB - EDP Sciences

VL - 46

IS - 5

SP - 1107

EP - 1120

AB - We examine an elliptic optimal control problem with control and state constraints in
ℝ3. An improved error estimate of
𝒪(hs)
with 3/4 ≤ s ≤ 1 − ε is proven for a discretisation
involving piecewise constant functions for the control and piecewise linear for the state.
The derived order of convergence is illustrated by a numerical example.

LA - eng

KW - Elliptic optimal control problem; state constraint; a priori error estimates; elliptic control problem; state constrained; a priori error estimates; convergence rate; finite elements

UR - http://eudml.org/doc/277841

ER -

## References

top- R.A. Adams and J.J.F. Fournier, Sobolev spaces. Academic Press, San Diego (2007).
- S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solution of elliptic partial differential equations satisfying general boundary conditions I. Comm. Pure Appl. Math.12 (1959) 623–727.
- J.W. Barrett and C.M. Elliott, A finite-element method for solving elliptic equations with Neumann data on a curved boundary using unfitted meshes. IMA J. Numer. Anal.4 (1984) 309–325.
- J. Bergh and J. Löfström, Interpolation spaces. Springer, Berlin (1976).
- M. Bergounioux, K. Ito and K. Kunisch, Primal-dual strategy for constrained optimal control problems. SIAM J. Control Optim.37 (1999) 1176–1194.
- E. Casas, Control of an elliptic problem with pointwise state constraints. SIAM J. Control Optim.4 (1986) 1309–1322.
- E. Casas and F. Tröltzsch, Recent advances in the analysis of pointwise state-constrained elliptic optimal control problems. ESAIM : COCV16 (2010) 581–600.
- S. Cherednichenko and A. Rösch, Error estimates for the regularization of optimal control problems with pointwise control and state constraints. Z. Anal. Anwendungen27 (2008) 195–212.
- S. Cherednichenko, K. Krumbiegel and A. Rösch, Error estimates for the Lavrentiev regularization of elliptic optimal control problems. Inverse Problems24 (2008).
- P.G. Ciarlet, The finite element method for elliptic problems. SIAM Classics In Applied Mathematics, Philadelphia (2002).
- J.C. de los Reyes, C. Meyer and B. Vexler, Finite element error analysis for state-constrained optimal control of the Stokes equations. Control and Cybernetics37 (2008) 251–284.
- K. Deckelnick and M. Hinze, Convergence of a finite element approximation to a state constrained elliptic control problem. SIAM J. Numer. Anal.45 (2007) 1937–1953.
- K. 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, edited by K. Kunisch, G. Of and O. Steinbach, Berlin, Heidelberg, Springer-Verlag (2008) 597–604.
- P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985).
- M. Hintermüller, K. Ito and K. Kunisch, The primal-dual active set strategy as a semi-smooth Newton method. SIAM J. Optim.13 (2003) 865–888.
- M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints. Springer-Verlag, Berlin (2009).
- K. Kunisch and A. Rösch, Primal-dual active set strategy for a general class of constrained optimal control problems. SIAM J. Optim.13 (2002) 321–334.
- C. Meyer, Error estimates for the finite-element approximation of an elliptic control problem with pointwise state and control constraints. Control and Cybernetics37 (2008) 51–85.
- C. Meyer, A. Rösch and F. Tröltzsch, Optimal control of PDEs with regularized pointwise state constraints. Comput. Optim. Appl.33 (2006) 209–228.
- P. Neittaanmaki, J. Sprekels and D. Tiba, Optimization of Elliptic Systems. Springer-Verlag, New York (2006).
- R. Rannacher, Zur L∞-Konvergenz linearer finiter elemente beim Dirichlet-problem. Math. Z.149 (1976) 69–77.
- A. Rösch and F. Tröltzsch, Existence of regular Lagrange multipliers for elliptic optimal control problem with pointwise control-state constraints. SIAM J. Optim.45 (2006) 548–564.
- A.H. Schatz, Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids. I : Global estimates. Math. Comput.67 (1998) 877–899.
- A.H. Schatz and L.B. Wahlbin, Interior maximum norm estimates for the finite element method. Math. Comput.31 (1977) 414–442.
- A.H. Schatz and L.B. Wahlbin, On the quasi-optimality in L∞ of the ${\stackrel{\u02da}{H}}^{1}$-projection into finite element spaces. Math. Comput.38 (1982) 1–22.
- A. Schmidt and K.G. Siebert, Design of Adaptive Finite Element Software, The Finite Element Toolbox ALBERTA. Springer-Verlag, Berlin (2000).
- F. Tröltzsch, Regular Lagrange multipliers for problems with pointwise mixed control-state constraints. SIAM J. Optim.15 (2005) 616–634.
- F. Tröltzsch, Optimal control of partial differential equations. Amer. Math. Soc., Providence, Rhode Island (2010).
- D.Z. Zanger, The inhomogeneous Neumann problem in Lipschitz domains. Commun. Partial Differ. Equ.25 (2000) 1771–1808.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.