Abstract
In this article we are concerned with the finite element discretization of optimal control problems subject to a second order elliptic PDE and additional pointwise constraints on the gradient of the state.
We will derive error estimates for the convergence of the cost functional under mesh refinement. Subsequently error estimates for the control and state variable are obtained.
As an intermediate tool we will also analyze a Moreau-Yosida regularized version of the optimal control problem. In particular we will derive convergence rates for the cost functional and the primal variables. To this end we will employ new techniques in estimating the L ∞-norm of the feasibility error which could also be used to improve existing estimates in the state constrained case.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
H. Blum, M. Dobrowolski, On finite element methods for elliptic equations on domains with corners. Computing 28, 53–63 (1982)
S. Brenner, L.R. Scott, The Mathematical Theory of Finite Element Methods, 3rd edn. (Springer, New York, 2008)
E. Casas, J.F. Bonnans, Contrôle de systèmes elliptiques semilinéares comportant des contraintes sur l’état, in Nonlinear Partial Differential Equations and their Applications 8, ed. by H. Brezzis, J.L. Lions (Longman, New York, 1988), pp. 69–86
E. Casas, L.A. Fernández, Optimal control of semilinear elliptic equations with pointwise constraints on the gradient of the state. Appl. Math. Optim. 27, 35–56 (1993)
K. Deckelnick, A. Günther, M. Hinze, Finite element approximation of elliptic control problems with constraints on the gradient. Numer. Math. 111, 335–350 (2008)
J. Douglas Jr., T. Dupont, L. Wahlbin, The stability in L q of the L 2-projection into finite element function spaces. Numer. Math. 23, 193–197 (1974/75)
P. Grisvard, Elliptic Problems in Nonsmooth Domains, 1st edn. Monographs and Studies in Mathematics (Pitman, Boston, 1985)
A. Günther, M. Hinze, Elliptic control problems with gradient constraints—variational discrete versus piecewise constant controls. Comput. Optim. Appl. 49(3), 549–566 (2009)
M. Hintermüller, M. Hinze, Moreau-Yosida regularization in state constrained elliptic control problems: error estimates and parameter adjustment. SIAM J. Numer. Anal. 47(3), 1666–1683 (2009)
M. Hintermüller, M. Hinze, R.H.W. Hoppe, Weak-duality based adaptive finite element methods for PDE-constrained optimization with pointwise gradient state-constraints. Preprint 2010-08, Hamburger Beiträge zur Angewandten Mathematik, 2010
M. Hintermüller, K. Kunisch, PDE-constrained optimization subject to pointwise constraints on the control, the state, and its derivative. SIAM J. Optim. 20(3), 1133–1156 (2009)
M. Hintermüller, A. Schiela, W. Wollner, The length of the primal-dual path in Moreau-Yosida-based path-following for state constrained optimal control. Preprint 2012-03, Hamburger Beiträge zur Angewandten Mathematik, 2012
M. Hinze, A variational discretization concept in control constrained optimization: the linear-quadratic case. Comp. Optim. Appl. 30(1), 45–61 (2005)
C. Ortner, W. Wollner, A priori error estimates for optimal control problems with pointwise constraints on the gradient of the state. Numer. Math. 118(3), 587–600 (2011)
A.H. Schatz, A weak discrete maximum principle and stability of the finite element method in L ∞ on plane polygonal domains. I. Math. Comp. 33(148), 77–91 (1980)
A.H. Schatz, L.B. Wahlbin, Maximum norm estimates in the finite element method on plane polygonal domains. Part 1. Math. Comp. 32(141), 73–109 (1978)
A. Schiela, W. Wollner, Barrier methods for optimal control problems with convex nonlinear gradient state constraints. SIAM J. Optim. 21(1), 269–286 (2011)
M. Ulbrich, Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems. MOS-SIAM Series on Optimization (SIAM, Philadelphia, 2011)
W. Wollner, A posteriori error estimates for a finite element discretization of interior point methods for an elliptic optimization problem with state constraints. Comput. Optim. Appl. 47(1), 133–159 (2010)
W. Wollner, Optimal control of elliptic equations with pointwise constraints on the gradient of the state in nonsmooth polygonal domains. SIAM J. Control Optim. 50(4), 2117–2129 (2012)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Basel
About this chapter
Cite this chapter
Wollner, W. (2013). A Priori Error Estimates for Optimal Control Problems with Constraints on the Gradient of the State on Nonsmooth Polygonal Domains. In: Bredies, K., Clason, C., Kunisch, K., von Winckel, G. (eds) Control and Optimization with PDE Constraints. International Series of Numerical Mathematics, vol 164. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0631-2_11
Download citation
DOI: https://doi.org/10.1007/978-3-0348-0631-2_11
Published:
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-0630-5
Online ISBN: 978-3-0348-0631-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)