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

ISBN-13: 9780817681487 / Angielski / Twarda / 2011 / 636 str.

Peter I. Kogut; Gunter Leugering

Opis

Optimal control of partial differential equations (PDEs) is by now, after more than 50 years of ever increasing scientific interest, 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, so that today fluid-structure interactions, magneto-hydromechanical, or electromagnetical as well as chemical and civil engineering problems can be dealt with. However, the numerical realization of optimal controls based on optimality conditions, together with the simulation of the states, 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 ever-increasingly complex 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 including lattice, honeycomb, and hierarchical structures. The investigation of optimal control problems for such structures is important to researchers working with cellular and hierarchical materials (lightweight materials) such as metallic and ceramic foams as well as bio-morphic material. Other modern engineering applications are chemical and civil engineering technologies, which often involve networked systems. Because of the complicated geometry of these structures--periodic media with holes or inclusions and a very small amount of material along layers or along bars--the asymptotic analysis is even more important, as a direct numerical computation of solutions would be extremely difficult. Specific topics 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, Optimal Control Problems for Partial Differential Equations on Reticulated Domains is an excellent reference-tool for graduate students, researchers, and practitioners in mathematics and areas of engineering involving reticulated domains.

Recenzje

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)

Spis treści

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.

Blurb

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.