**Graduate Studies in Mathematics**

Volume: 112;
2010;
399 pp;
Hardcover

MSC: Primary 49; 35; 90;

Print ISBN: 978-0-8218-4904-0

Product Code: GSM/112

List Price: $78.00

AMS Member Price: $62.40

MAA Member Price: $70.20

**Electronic ISBN: 978-1-4704-1174-9
Product Code: GSM/112.E**

List Price: $73.00

AMS Member Price: $58.40

MAA Member Price: $65.70

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#### Supplemental Materials

# Optimal Control of Partial Differential Equations: Theory, Methods and Applications

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*Fredi Tröltzsch*

Translated by Jürgen Sprekels

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, Tröltzsch 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.

#### Readership

Graduate students and research mathematicians interested in optimal control theory and PDEs.

#### Reviews & Endorsements

The book provides a thorough and self-contained introduction...[It includes] carefully chosen examples...The presentation of the material is clear and self-contained. A great deal of attention is paid to careful exposition of relevant supporting tools from nonlinear analysis and PDEs. ...A wealth of examples... [T]his is a very carefully written text with an eye on graduate students wishing to enter the field of PDE optimal control. The material presented is fairly complete, self-contained and well exposed.

-- Irena Lasiecka, Mathematical Reviews

#### Table of Contents

# Table of Contents

## Optimal Control of Partial Differential Equations: Theory, Methods and Applications

- Cover Cover11 free
- Title page iii4 free
- Contents vii8 free
- Preface xi12 free
- Introduction and examples 118 free
- Linear-quadratic elliptic control problems 2138 free
- Linear-quadratic parabolic control problems 119136
- Optimal control of semilinear elliptic equations 181198
- Optimal control of semilienar parabolic equations 265282
- Optimization problems in Banach spaces 323340
- Supplementary results on partial differential equations 355372
- Bibliography 385402
- Index 397414 free
- Back Cover Back Cover1418