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Product Code:  GSM/112 
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Electronic ISBN:  9781470411749 
Product Code:  GSM/112.E 
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Book DetailsGraduate Studies in MathematicsVolume: 112; 2010; 399 ppMSC: Primary 49; 35; 90;
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, secondorder sufficient conditions, and main principles of selected numerical techniques. It also contains a survey on the KarushKuhnTucker 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 selfcontained, 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 selfcontained 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 PDEconstrained optimization.ReadershipGraduate students and research mathematicians interested in optimal control theory and PDEs.

Table of Contents

Chapters

Chapter 1. Introduction and examples

Chapter 2. Linearquadratic elliptic control problems

Chapter 3. Linearquadratic parabolic control problems

Chapter 4. Optimal control of semilinear elliptic equations

Chapter 5. Optimal control of semilinear parabolic equations

Chapter 6. Optimization problems in Banach spaces

Chapter 7. Supplementary results on partial differential equations


Additional Material

Reviews

The book provides a thorough and selfcontained introduction...[It includes] carefully chosen examples...The presentation of the material is clear and selfcontained. 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, selfcontained and well exposed.
Irena Lasiecka, Mathematical Reviews


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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, secondorder sufficient conditions, and main principles of selected numerical techniques. It also contains a survey on the KarushKuhnTucker 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 selfcontained, 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 selfcontained 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 PDEconstrained optimization.
Graduate students and research mathematicians interested in optimal control theory and PDEs.

Chapters

Chapter 1. Introduction and examples

Chapter 2. Linearquadratic elliptic control problems

Chapter 3. Linearquadratic parabolic control problems

Chapter 4. Optimal control of semilinear elliptic equations

Chapter 5. Optimal control of semilinear parabolic equations

Chapter 6. Optimization problems in Banach spaces

Chapter 7. Supplementary results on partial differential equations

The book provides a thorough and selfcontained introduction...[It includes] carefully chosen examples...The presentation of the material is clear and selfcontained. 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, selfcontained and well exposed.
Irena Lasiecka, Mathematical Reviews