Synopses & Reviews
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.
Review
From the reviews: "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)
Synopsis
In the development of optimal control, the complexity of the systems
Synopsis
This book offers a method based on asymptotic analysis aimed at combining techniques of homogenization and approximation to cover optimal control problems defined on reticulated domains, networked systems such as lattice, honeycomb, or hierarchical structures.
Synopsis
After over 50 years of increasing scientific interest, optimal control of partial differential equations
Synopsis
In the development of optimal control, the complexity of the systems
Synopsis
In the development of optimal control, the complexity of the systems
Table 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