Optimal Control Problems for Partial Differential Equations on Reticulated Domains - Approximation and Asymptotic Analysis

Optimal control of partial differential equations (PDEs) is a well-established discipline in mathematics with many interfaces to science and engineering. During the development of this area, the complexity of the systems to be controlled has also increased significantly; however, the numerical realization of these complex systems has become an issue in scientific computing, as the number of variables involved may easily exceed a couple of million.
In order to carry out model-reduction on these systems, the authors of this work have developed a method based on asymptotic analysis. They aim at combining techniques of homogenization and approximation in order to cover optimal control problems defined on reticulated domains, networked systems such as lattice, honeycomb, or hierarchical structures. Because of these structures' complicated geometry, the asymptotic analysis is even more important, as a direct numerical computation of solutions would be extremely difficult. The work's first part can be used as an advanced textbook on abstract optimal control problems, in particular on reticulated domains, while the second part serves as a research monograph where stratified applications are discussed.
Optimal Control Problems for Partial Differential Equations on Reticulated Domains is an excellent reference for graduate students, researchers, and practitioners in mathematics and areas of engineering involving reticulated domains and networked systems.

List of contents

Introduction.- Part I. Asymptotic Analysis of Optimal Control Problems for Partial Differential Equations: Basic Tools.- Background Material on Asymptotic Analysis of External Problems.- Variational Methods of Optimal Control Theory.- Suboptimal and Approximate Solutions to External Problems.- Introduction to the Asymptotic Analysis of Optimal Control Problems: A Parade of Examples.- Convergence Concepts in Variable Banach Spaces.- Convergence of Sets in Variable Spaces.- Passing to the Limit in Constrained Minimization Problems.- Part II. Optimal Control Problems on Periodic Reticulated Domains: Asymptotic Analysis and Approximate Solutions.- Suboptimal Control of Linear Steady-States Processes on Thin Periodic Structures with Mixed Boundary Controls.- Approximate Solutions of Optimal Control Problems for Ill-Posed Objects on Thin Periodic Structures.- Asymptotic Analysis of Optimal Control Problems on Periodic Singular Structures.- Suboptimal Boundary Control of Elliptic Equations in Domains with Small Holes.- Asymptotic Analysis of Elliptic Optimal Control Problems in Thick Multi-Structures with Dirichlet and Neumann Boundary Controls.- Gap Phenomenon in Modeling of Suboptimal Controls to Parabolic Optimal Control Problems in Thick Multi-Structures.- Boundary Velocity Suboptimal Control of Incompressible Flow in Cylindrically Perforated Domains.- Optimal Control Problems in Coefficients: Sensitivity Analysis and Approximation.- References.- Index.

Summary

Optimal control of partial differential equations (PDEs) is a well-established discipline in mathematics with many interfaces to science and engineering. During the development of this area, the complexity of the systems to be controlled has also increased significantly; however, the numerical realization of these complex systems has become an issue in scientific computing, as the number of variables involved may easily exceed a couple of million.
In order to carry out model-reduction on these systems, the authors of this work have developed a method based on asymptotic analysis. They aim at combining techniques of homogenization and approximation in order to cover optimal control problems defined on reticulated domains, networked systems such as lattice, honeycomb, or hierarchical structures. Because of these structures' complicated geometry, the asymptotic analysis is even more important, as a direct numerical computation of solutions would be extremely difficult. The work's first part can be used as an advanced textbook on abstract optimal control problems, in particular on reticulated domains, while the second part serves as a research monograph where stratified applications are discussed.
Optimal Control Problems for Partial Differential Equations on Reticulated Domains is an excellent reference for graduate students, researchers, and practitioners in mathematics and areas of engineering involving reticulated domains and networked systems.

Additional text

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)

Report

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)

Product details

• Authors: Peter Kogut, Peter I Kogut, Peter I. Kogut, G. Nter Leugering, Gunter Leugering, Günter Leugering, Günter R Leugering, Gunter R. Leugering, Günter R. Leugering
• Publisher: Springer, Basel
• Languages: English
• Product format: Hardback
• Released: 16.12.2010
• EAN: 9780817681487
• ISBN: 978-0-8176-8148-7
• No. of pages: 636
• Dimensions: 165 mm x 241 mm x 40 mm
• Weight: 1117 g
• Illustrations: XVI, 636 p. 26 illus.
• Series: Systems & Control: Foundations & Applications; Systems & Control: Foundations & Applications; Systems & Control: Foundations; Systems& Control: Foundations & Applications
• Subject: Natural sciences, medicine, IT, technology > Mathematics > Analysis