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 reticulatedstructures.
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.
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)
