Systems & Control: Foundations & Applications

Optimal Control Problems for Partial Differential Equations on Reticulated Domains

Approximation and Asymptotic Analysis

Authors: Kogut, Peter I., Leugering, Günter R.

  • Applications from mechanics and material sciences have been simplified so that prerequisite knowledge in those areas is not required
  • Methods described are useful for researchers working with reticulated—or networked—systems, including chemical and civil engineers and materials scientists
  • Synthesizes previous results of the authors with those of scientists across the discipline to provide a fresh look at the subject
  • Can be used as an advanced textbook on abstract optimal control problems, while the second part of the book serves as a research monograph where stratified applications are discussed
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About this book

After over 50 years of increasing scientific interest, optimal control of partial differential equations (PDEs) has developed into a well-established discipline in mathematics with myriad applications to science and engineering. As the field has grown, so too has the complexity of the systems it describes; the numerical realization of optimal controls has become increasingly difficult, demanding ever more sophisticated mathematical tools. 

A comprehensive monograph on the subject, Optimal Control of Partial Differential Equations on Reticulated Domains is intended to address some of the obstacles that face researchers today, particularly with regard to multi-scale engineering applications involving hierarchies of grid-like domains. Bringing original results together with others previously scattered across the literature, it tackles computational challenges by exploiting asymptotic analysis and harnessing differences between optimal control problems and their underlying PDEs.

The book consists of two parts, the first of which can be viewed as a compendium of modern optimal control theory in Banach spaces. The second part is a focused, in-depth, and self-contained study of the asymptotics of optimal control problems related to reticulated domains—the first such study in the literature. Specific topics covered in the work include:

* a mostly self-contained mathematical theory of PDEs on reticulated domains;

* the concept of optimal control problems for PDEs in varying such domains, and hence, in varying Banach spaces;

* convergence of optimal control problems in variable spaces;

* an introduction to the asymptotic analysis of optimal control problems;

* optimal control problems dealing with ill-posed objects on thin periodic structures, thick periodic singular graphs, thick multi-structures with Dirichlet and Neumann boundary controls, and coefficients on reticulated structures.

Serving as both a text on abstract optimal control problems and a monograph where specific applications are explored, this book is an excellent reference for graduate students, researchers, and practitioners in mathematics and areas of engineering involving reticulated domains.

Reviews

From the reviews:

“The book under review aims to introduce the reader to various classes of optimal control problems (briefly OCP) governed by partial differential equations and to several applications to problems in engineering that can be modeled by them. … The book is very well conceived and the material is organized in a clear and complete way, starting from basic tools such as measure theory, Sobolev spaces, functional analysis, and general variational problems.” (Giuseppe Buttazzo, Mathematical Reviews, August, 2013)

“This book introduces in the mathematical world of optimal control problems posed in reticulated domains. … a great number of very nice and well written examples illustrate the main difficulties behind the questions and the reasons for posing them. The book provides a very good introduction into this important topic and may serve as the basis for a one semester course on optimal control in reticulated domains and for an associated seminary, where specific aspects of the theory can be discussed.” (Fredi Tröltzsch, Zentralblatt MATH, Vol. 1253, 2013)


Table of contents (16 chapters)

  • Introduction

    Kogut, Peter I. (et al.)

    Pages 1-11

  • Background Material on Asymptotic Analysis of Extremal Problems

    Kogut, Peter I. (et al.)

    Pages 15-61

  • Variational Methods of Optimal Control Theory

    Kogut, Peter I. (et al.)

    Pages 63-111

  • Suboptimal and Approximate Solutions to Extremal Problems

    Kogut, Peter I. (et al.)

    Pages 113-132

  • Introduction to the Asymptotic Analysis of Optimal Control Problems: A Parade of Examples

    Kogut, Peter I. (et al.)

    Pages 133-159

Buy this book

eBook $139.00
price for USA (gross)
  • ISBN 978-0-8176-8149-4
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $179.00
price for USA
  • ISBN 978-0-8176-8148-7
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Rent the ebook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
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Bibliographic Information

Bibliographic Information
Book Title
Optimal Control Problems for Partial Differential Equations on Reticulated Domains
Book Subtitle
Approximation and Asymptotic Analysis
Authors
Series Title
Systems & Control: Foundations & Applications
Copyright
2011
Publisher
Birkhäuser Basel
Copyright Holder
Springer Science+Business Media, LLC
eBook ISBN
978-0-8176-8149-4
DOI
10.1007/978-0-8176-8149-4
Hardcover ISBN
978-0-8176-8148-7
Series ISSN
2324-9749
Edition Number
1
Number of Pages
XVI, 636
Number of Illustrations and Tables
26 b/w illustrations
Topics