Relaxation in Optimization Theory and Variational Calculus (De Gruyter Series in Nonlinear Analysis & Applications) (De Gruyter Series in Nonlinear Analysis & Applications, 4) - Hardcover

Roubicek, Tomas

Hardcover

ISBN 10: 3110145421 ISBN 13: 9783110145427

Publisher: De Gruyter, 1997

Synopsis

The series is devoted to the publication of high-level monographs which cover the whole spectrum of current nonlinear analysis and applications in various fields, such as optimization, control theory, systems theory, mechanics, engineering, and other sciences. One of its main objectives is to make available to the professional community expositions of results and foundations of methods that play an important role in both the theory and applications of nonlinear analysis. Contributions which are on the borderline of nonlinear analysis and related fields and which stimulate further research at the crossroads of these areas are particularly welcome. Editor-in-Chief Jürgen Appell, Würzburg, Germany Honorary and Advisory Editors Catherine Bandle, Basel, Switzerland Alain Bensoussan, Richardson, Texas, USA Avner Friedman, Columbus, Ohio, USA Umberto Mosco, Worcester, Massachusetts, USA Editorial Board Manuel del Pino, Bath, UK, and Santiago, Chile Mikio Kato, Nagano, Japan Wojciech Kryszewski, Toruń, Poland Vicenţiu D. Rădulescu, Kraków, Poland Simeon Reich, Haifa, Israel Please submit book proposals to Jürgen Appell. Titles in planning include Ireneo Peral Alonso and Fernando Soria, Elliptic and Parabolic Equations Involving the Hardy–Leray Potential (2020) Cyril Tintarev, Profile Decompositions and Cocompactness: Functional-Analytic Theory of Concentration Compactness (2020) Takashi Suzuki, Semilinear Elliptic Equations: Classical and Modern Theories (2021)

About this title

Synopsis

The purpose of this book is to introduce the reader to a relaxation method based on a continuous extension of various optimization problems on so-called convex compactifications, which proves useful when dealing particularly with problems in optimal control theory, calculus of variations, and non-cooperative game theory. In the first two chapters background results are summarized, and the general theory of convex compactifications developed by the author is introduced. Later in this theory is used to obtain convex locally compact envelopes of the Lebesgue and Sobolev spaces involved in concrete problems From a unified point of view, these nontrivial envelopes cover the classical Young measures as well as various generalizations of them, which can record the 'limit behaviour' of fast oscillation and concentration effects. Existence and stability of the generalized solutions are then guaranteed by compactness, while convexity allows the optimality conditions resulting in Pontryagin and Weirstrab maximum principles to be formulated.

This is demonstrated by optimal control problems governed by ordinary or partial differential equations and integral equations, as well as by scalar or vectorial multi-dimensional variational problems. For problems in game theory, both compactness and convexity are needed to obtain existence results (in terms of mixed strategies). Approximation theory and numerical results of sample examples are also presented.

Edition details

• Publisher: De Gruyter
• Publication date: 1997
• Language: English
• ISBN 10: 3110145421
• ISBN 13: 9783110145427
• Binding: Hardcover
• Number of pages: 488