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A Goal-Oriented Adaptive Moreau-Yosida Algorithm for Control- and State-Constrained Elliptic Control Problems

Part of: Optimality conditions Partial differential equations, boundary value problems Existence theories

Published online by Cambridge University Press:  27 January 2016

Andreas Günther  and
Moulay Hicham Tber
Andreas Günther
Affiliation:
Bereich Optimierung und Approximation, Universität Hamburg, Bundesstraße 55, 20146 Hamburg, Germany
Moulay Hicham Tber*
Affiliation:
Cadi Ayyad University, Av. Abdelkrim Khattabi, B.P. 511–40000–Marrakech, Morocco
*
*Corresponding author. Email:a.guenther@uni-hamburg.de (A. Günther), hicham.tber@gmail.com (M. H. Tber)
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Abstract

In this work, we develop an adaptive algorithm for solving elliptic optimal control problems with simultaneously appearing state and control constraints. The algorithm combines a Moreau-Yosida technique for handling state constraints with a semi-smooth Newton method for solving the optimality systems of the regularized sub-problems. The state and adjoint variables are discretized using continuous piecewise linear finite elements while a variational discretization concept is applied for the control. To perform the adaptive mesh refinements cycle we derive local error estimators which extend the goal-oriented error approach to our setting. The performance of the overall adaptive solver is assessed by numerical examples.

Keywords

Elliptic optimal control problemcontrol and state constraintsMoreau-Yosida regularizationsemi-smooth Newton methodvariational discretizationgoal-oriented adaptivity

MSC classification

Secondary: 49J20: Optimal control problems involving partial differential equations 49K20: Problems involving partial differential equations 65K10: Optimization and variational techniques 65N50: Mesh generation and refinement
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 8 , Issue 3 , June 2016 , pp. 426 - 448
Copyright
Copyright © Global-Science Press 2016 

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