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Generalized penalty methods for elliptic Neumann boundary control problem with state and control constraints

Authors
Publisher
Technische Universiteit Eindhoven
Publication Date
Keywords
  • Mathematical Optimization (General)
  • Numerical Solution Of Partial Differental Equations
  • Mathematical Optimisation
  • Partial Differential Equations
Disciplines
  • Mathematics

Abstract

Technisch-Naturwissenschaftliche Fakulta¨t JOHANNES KEPLER UN IVERS I T A¨T L INZ Ne t zw e r k f u¨ r F o r s c h u n g , L e h r e u n d P r a x i s Generalized Penalty Methods for Elliptic Neumann Boundary Control Problem with State and Control Constraints MASTERARBERIT zur Erlangung des akademischen Grades Diplom-Ingenieur in der Studienrichtung INDUSTRIAL MATHEMATICS Angefertigt am Institut fu¨r Numerische Mathematik Betreuung: O.Univ. Prof. Dipl. Ing. Dr. Helmut Gfrerer Eingereicht von: Esubalewe Lakie Yedeg Linz, July 2010 Johannes Kepler Universita¨t A-4040 Linz · Altenbergerstraße 69 · Internet: http://www.jku.at · DVR 0093696 Dedicated to Tigist Muluken and Wubit Aragaw Abstract In this work we studied an optimal control problem constrained with boundary control. It is an infinite dimensional convex optimization problem that consist of minimizing a cost function subject to pointwise state and control constraints and governed by elliptic differential equation with a Neumann boundary condition. The control being distributed only on the boundary. A generalized penalty function approach is then used to reformulate the original constrained problem as an unconstrained problem. The convergence results and the error estimates of the penalty method are stated. To solve the resulting subproblems numerically we used Newton’s method with line search in function spaces and its superlinear convergence is presented. The finite element method is used to discretize the subproblems and to transform into finite dimensional problems. Finally, numerical examples are given to illustrate the theoretical results. i Acknowledgement First of all, I would like to thank my advisor Professor Helmut Gfrerer, for supervising my thesis, and the countless discussions, guidance and inspirations throughout this work. I owe my deepest gratitude to him for making the thesis possible. I would like to take this moment to thank my program

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