Optimization of Elliptic Systems: Theory and Applications

ISBN-13: 9780387272351 / Angielski / Twarda / 2005 / 512 str.

Pekka Neittaanmaki; J]rgen Sprekels; Dan Tiba

Opis

The present monograph is intended to provide a comprehensive and accessible introduction to the optimization of elliptic systems. This area of mathematical research, which has many important applications in science and technology. has experienced an impressive development during the past two decades. There are already many good textbooks dealing with various aspects of optimal design problems. In this regard, we refer to the works of Pironneau 1984], Haslinger and Neittaanmaki 1988], 1996], Sokolowski and Zolksio 1992], Litvinov 2000], Allaire 2001], Mohammadi and Pironneau 2001], Delfour and Zolksio 2001], and Makinen and Haslinger 2003]. Already Lions I9681 devoted a major part of his classical monograph on the optimal control of partial differential equations to the optimization of elliptic systems. Let us also mention that even the very first known problem of the calculus of variations, the brachistochrone studied by Bernoulli back in 1696. is in fact a shape optimization problem. The natural richness of this mathematical research subject, as well as the extremely large field of possible applications, has created the unusual situation that although many important results and methods have already been est- lished, there are still pressing unsolved questions. In this monograph, we aim to address some of these open problems; as a consequence, there is only a minor overlap with the textbooks already existing in the field.

Recenzje

From the reviews:

"The book gives a comprehensive view of the optimization of systems governed by partial differential equations of elliptic type. ... The book is carefully written. Basic tools of convex analysis, an abstract optimization theory, notions on the well-posedness of elliptic systems, and the existence results for optimal control problems are presented in a simple way. ... The book will therefore be useful for confirmed researchers as well as students willing to enter in the field. One could even teach a graduate course by selecting material." (Joseph Frédéric Bonnans, Zentralblatt MATH, Vol. 1106 (8), 2007)

Spis treści

Introductory Topics.- Existence.- Optimality Conditions.- Discretization.- Unknown Domains.- Optimization of Curved Mechanical Systems.

Blurb

This monograph provides a comprehensive and accessible introduction to the optimization of elliptic systems. This area of mathematical research, which has many important application in science and technology, has experienced an impressive development during the last two decades. This monograph aims to address some of the pressing unsolved questions in the field. The exposition concentrates along two main directions: the optimal control of linear and nonlinear elliptic equations, and problems involving unknown and/or variable domains. Throughout this monograph, the authors elucidate connections between seemingly different types of problems. One basic feature is to relax the needed regularity assumptions as much as possible in order to include larger classes of possible applications. The book is organized into six chapters that give a gradual and accessible presentation of the material, and a special effort is made to present numerous examples.

This monograph is addressed primarily to mathematics graduate students and researchers, however much of this material will also prove useful for scientists from physics, mechanics, and engineering.