Fredi Troltzsch – författare

Optimal control theory is concerned with finding control functions that minimize cost functions for systems described by differential equations. The methods have found widespread applications in aeronautics, mechanical engineering, the life sciences, and many other disciplines. This book focuses on optimal control problems where the state equation is an elliptic or parabolic partial differential equation. Included are topics such as the existence of optimal solutions, necessary optimality conditions and adjoint equations, second-order sufficient conditions, and main principles of selected numerical techniques. It also contains a survey on the Karush-Kuhn-Tucker theory of nonlinear programming in Banach spaces. The exposition begins with control problems with linear equations, quadratic cost functions and control constraints. To make the book self-contained, basic facts on weak solutions of elliptic and parabolic equations are introduced. Principles of functional analysis are introduced and explained as they are needed. Many simple examples illustrate the theory and its hidden difficulties. This start to the book makes it fairly self-contained and suitable for advanced undergraduates or beginning graduate students. Advanced control problems for nonlinear partial differential equations are also discussed. As prerequisites, results on boundedness and continuity of solutions to semilinear elliptic and parabolic equations are addressed. These topics are not yet readily available in books on PDEs, making the exposition also interesting for researchers. Alongside the main theme of the analysis of problems of optimal control, Troltzsch also discusses numerical techniques. The exposition is confined to brief introductions into the basic ideas in order to give the reader an impression of how the theory can be realized numerically. After reading this book, the reader will be familiar with the main principles of the numerical analysis of PDE-constrained optimization.

Interest in the area of control of systems defined by partial differential Equations has increased strongly in recent years. A major reason has been the requirement of these systems for sensible continuum mechanical modelling and optimization or control techniques which account for typical physical phenomena. Particular examples of problems on which substantial progress has been made are the control and stabilization of mechatronic structures, the control of growth of thin films and crystals, the control of Laser and semi-conductor devices, and shape optimization problems for turbomachine blades, shells, smart materials and microdiffractive optics. This volume contains original articles by world reknowned experts in the fields of optimal control of partial differential equations, shape optimization, numerical methods for partial differential equations and fluid dynamics, all of whom have contributed to the analysis and solution of many of the problems discussed. The collection provides a state-of-the-art overview of the most challenging and exciting recent developments in the field. It is geared towards postgraduate students and researchers dealing with the theoretical and practical aspects of a wide variety of high technology problems in applied mathematics, fluid control, optimal design, and computer modelling.

Die mathematische Optimierung von Vorgängen, die durch partielle Differentialgleichun­ gen modelliert werden, hat in den letzten Jahren einen beachtlichen Aufschwung ge­ nommen. Die Verfügbarkeit immer besserer Computer ermöglichte neue interessante An­ wendungen dieses Gebietes in der Praxis, etwa in Strömungsmechanik, Mikroelektronik, Kristallzüchtung, Gefäßchirurgie oder Herzmedizin, um nur einige Beispiele zu nennen. Im Zusammenhang damit ist das Interesse von Numerikern und Optimierern an der Ver­ wendung ihrer Methoden in der Optimalsteuerung von partiellen Differentialgleichungen deutlich gestiegen und es gibt mehr Nachfrage von Studenten und Doktoranden der Ma­ thematik nach einer Einführung in Grundideen der zugehörigen Theorie. Heute existiert eine Reihe von Monographien zu verschiedenen Aspekten der optimalen Steuerung von partiellen Differentialgleichungen, insbesondere das bekannte Standard­ werk von J.L. Lions [121], dem für Aufgaben mit linearen Gleichungen und konvexen Zielfunktionalen kaum etwas hinzuzufügen ist. Das Interesse am Skript meiner Vorlesun­ gen zur optimalen Steuerung an den technischen Universitäten in Chemnitz und Berlin zeigte jedoch den Bedarf an einer Einführung in die Thematik in deutscher Sprache, die auch Aspekte der nichtlinearen Optimierung im Funktionenraum berücksichtigt.

This book is a collection of thoroughly refereed papers presented at the 25th IFIP TC 7 Conference on System Modeling and Optimization, held in Dresden, Germany, in September 2011. The 55 revised papers were carefully selected from numerous submissions. They are organized in the following topical sections: control of distributed parameter systems; stochastic optimization and control; stabilization, feedback, and model predictive control; flow control; shape and structural optimization; and applications and control of lumped parameter systems.

Along with the dramatic increase in computer power, the application of PDE-based control theory and the corresponding numerical algorithms to industrial problems has become increasingly important. This development is reflected in this volume. The mathematical framework contains modelling and analysis of such systems as well as the numerical analysis and implementation.