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This book presents a systematic approach to the numerical analysis of several carefully selected classes of optimal control problems governed by elliptic partial differential equations (PDEs). A priori error estimates for the discretization error between the optimal solutions of the continuous and discretized problems are derived, and numerical experiments are included to illustrate the results.
The proofs are presented in a structured and accessible manner, facilitating a clear understanding of the techniques used. The necessary results from functional analysis and elliptic PDE theory are provided in a self-contained way. Essential aspects of finite element theory including some results newly established by the author(s) are also covered, with all relevant proofs provided in full detail.
Portions of the material have been successfully used in graduate-level courses and seminars, as well as in Master s and PhD theses. The book is intended for researchers and graduate students interested in finite element analysis and/or optimal control problems involving PDEs.
List of contents
Chapter 1. Introduction.- Part I. Theory and Finite Elements for Elliptic PDEs.- Chapter 2. Topics from the Theory of Elliptic PDEs.- Chapter 3. Topics from Finite Elements for Elliptic PDEs.- Part II. Distributed Control Problems.- Chapter 4. No Inequality Constraints.- Chapter 5. Control Constraints.- Chapter 6. Bang-Bang Controls.- Chapter 7. Semilinear State Equation.- Part III. Boundary Control Problems.- Chapter 8. Neumann Control.- Chapter 9. Dirichlet Control.- Part IV. Problems Involving Dirac Measures.- Chapter 10. Pointwise Control.- Chapter 11. Pointwise Tracking.- Chapter 12. Finitely Many Pointwise State Constraints.- Part V. Problems Involving General Measures.- Chapter 13. State Constraints.- Chapter 14. Sparse Controls.
About the author
Dominik Meidner studied Mathematics with a minor in Computer Science at the Ruprecht-Karls-University Heidelberg, graduating with distinction in 2003. From 2003 to 2008, Dr. Meidner pursued his doctorate at the same university, completing his dissertation on adaptive space-time finite element methods for optimization problems governed by nonlinear parabolic systems with summa cum laude. Dr. Meidner began his career as a research assistant at the Ruprecht-Karls-University Heidelberg and moved to the Technical University of Munich (TUM) in 2008. Since 2019, he has been a Senior Lecturer (Akademischer Oberrat) at the Chair of Optimal Control at TUM. He has published numerous scientific papers and received several awards. Dr. Meidner is also involved in software development and has contributed to the finite element software "Gascoigne" and the C++ optimization library "RoDoBo."
Boris Vexler's research area is the numerical analysis of problems described by partial differential equations (PDEs) with a special focus on optimization problems with PDEs. He studied at Lomonosov University in Moscow and the University of Heidelberg. He obtained his doctorate at Heidelberg in 2004 and his lecturer qualification (Habilitation) at the University of Graz in 2008. After completing his doctorate, he worked at the RICAM Institute of the Austrian Academy of Sciences in Linz and, in 2008, was appointed professor for control theory at TUM. After rejecting calls to the universities of Vienna and Düsseldorf, Prof. Vexler was appointed to the Chair of Optimal Control at TUM in 2013. He served as speaker of the International Research Training Group IGDK 1754 from 2012 to 2022, and from 2015 to 2018, he was the dean of studies of the TUM Department of Mathematics.
Summary
This book presents a systematic approach to the numerical analysis of several carefully selected classes of optimal control problems governed by elliptic partial differential equations (PDEs). A priori error estimates for the discretization error between the optimal solutions of the continuous and discretized problems are derived, and numerical experiments are included to illustrate the results.
The proofs are presented in a structured and accessible manner, facilitating a clear understanding of the techniques used. The necessary results from functional analysis and elliptic PDE theory are provided in a self-contained way. Essential aspects of finite element theory—including some results newly established by the author(s)—are also covered, with all relevant proofs provided in full detail.
Portions of the material have been successfully used in graduate-level courses and seminars, as well as in Master’s and PhD theses. The book is intended for researchers and graduate students interested in finite element analysis and/or optimal control problems involving PDEs.
