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Control and Optimization with PDE Constraints pp 193–215Cite as

Birkhäuser

A Priori Error Estimates for Optimal Control Problems with Constraints on the Gradient of the State on Nonsmooth Polygonal Domains

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Part of the book series: International Series of Numerical Mathematics ((ISNM,volume 164))

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.

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Authors and Affiliations

  1. Department of Mathematics, University of Hamburg, Bundesstrasse 55, D-20146, Hamburg, Germany

    Winnifried Wollner

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  1. Winnifried Wollner
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Correspondence to Winnifried Wollner .

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Editors and Affiliations

  1. , Institute for Mathematics and Scientific, University of Graz, Heinrichstr. 36, Graz, 8010, Austria

    Kristian Bredies

  2. , Institute for Mathematics and Scientific, University of Graz, Heinrichstr. 36, Graz, 8010, Austria

    Christian Clason

  3. , Institute for Mathematics and Scientific, University of Graz, Heinrichstr. 36, Graz, 8010, Austria

    Karl Kunisch  & Gregory von Winckel  & 

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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

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