Young measures are now a widely used tool in the Calculus of Variations, in Control Theory, in Probability Theory and other fields. They are known under different names such as "relaxed controls", "fuzzy random variables" and many other names. This monograph provides a unified presentation of the theory, along with new results and applications in various fields. It can serve as a reference on the subject. Young measures are presented in a general setting which includes finite and for the first time infinite dimensional spaces: the fields of applications of Young measures (Control Theory, Calculus of Variations, Probability Theory...) are often concerned with problems in infinite dimensional settings. The theory of Young measures is now well understood in a finite dimensional setting, but open problems remain in the infinite dimensional case. We provide several new results in the general frame, which are new even in the finite dimensional setting, such as characterizations of convergence in measure of Young measures (Chapter 3) and compactness criteria (Chapter 4).These results are established under a different form (and with fewer details and developments) in recent papers by the same authors. We also provide new applications to Visintin and Reshetnyak type theorems (Chapters 6 and 8), existence of solutions to differential inclusions (Chapter 7), dynamical programming (Chapter 8) and the Central Limit Theorem in locally convex spaces (Chapter 9).

In recent years, technological progress created a great need for complex mathematical models. Many practical problems can be formulated using optimization theory and they hope to obtain an optimal solution. In most cases, such optimal solution can not be found. So, non-convex optimization problems (arising, e.g., in variational calculus, optimal control, nonlinear evolutions equations) may not possess a classical minimizer because the minimizing sequences have typically rapid oscillations. This behavior requires a relaxation of notion of solution for such problems; often we can obtain such a relaxation by means of Young measures. This monograph is a self-contained book which gathers all theoretical aspects related to the defining of Young measures (measurability, disintegration, stable convergence, compactness), a book which is also a useful tool for those interested in theoretical foundations of the measure theory. It provides a complete set of classical and recent compactness results in measure and function spaces. The book is organized in three chapters: The first chapter covers background material on measure theory in abstract frame. In the second chapter the measure theory on topological spaces is presented. Compactness results from the first two chapters are used to study Young measures in the third chapter. All results are accompanied by full demonstrations and for many of these results different proofs are given. All statements are fully justified and proved.

This book provides a thorough exposition of the main concepts and results related to various types of convergence of measures arising in measure theory, probability theory, functional analysis, partial differential equations, mathematical physics, and other theoretical and applied fields. Particular attention is given to weak convergence of measures. The principal material is oriented toward a broad circle of readers dealing with convergence in distribution of random variables and weak convergence of measures. The book contains the necessary background from measure theory and functional analysis. Large complementary sections aimed at researchers present the most important recent achievements. More than 100 exercises (ranging from easy introductory exercises to rather difficult problems for experienced readers) are given with hints, solutions, or references. Historic and bibliographic comments are included. The target readership includes mathematicians and physicists whose research is related to probability theory, mathematical statistics, functional analysis, and mathematical physics.

This book provides a comprehensive guide to analyzing and solving optimal design problems in continuous media by means of the so-called sub-relaxation method. Though the underlying ideas are borrowed from other, more classical approaches, here they are used and organized in a novel way, yielding a distinct perspective on how to approach this kind of optimization problems. Starting with a discussion of the background motivation, the book broadly explains the sub-relaxation method in general terms, helping readers to grasp, from the very beginning, the driving idea and where the text is heading. In addition to the analytical content of the method, it examines practical issues like optimality and numerical approximation. Though the primary focus is on the development of the method for the conductivity context, the book’s final two chapters explore several extensions of the method to other problems, as well as formal proofs. The text can be used for a graduate course in optimal design, even if the method would require some familiarity with the main analytical issues associated with this type of problems. This can be addressed with the help of the provided bibliography.

Robust design—that is, managing design uncertainties such as model uncertainty or parametric uncertainty—is the often unpleasant issue crucial in much multidisciplinary optimal design work. Recently, there has been enormous practical interest in strategies for applying optimization tools to the development of robust solutions and designs in several areas, including aerodynamics, the integration of sensing (e.g., laser radars, vision-based systems, and millimeter-wave radars) and control, cooperative control with poorly modeled uncertainty, cascading failures in military and civilian applications, multi-mode seekers/sensor fusion, and data association problems and tracking systems. The contributions to this book explore these different strategies. The expression "optimization-directed” in this book’s title is meant to suggest that the focus is not agonizing over whether optimization strategies identify a true global optimum, but rather whether these strategies make significant design improvements.

A lot of economic problems can be formulated as constrained optimizations and equilibration of their solutions. Various mathematical theories have been supplying economists with indispensable machineries for these problems arising in economic theory. Conversely, mathematicians have been stimulated by various mathematical difficulties raised by economic theories. The series is designed to bring together those mathematicians who are seriously interested in getting new challenging stimuli from economic theories with those economists who are seeking effective mathematical tools for their research.

The series is designed to bring together those mathematicians who are seriously interested in getting new challenging stimuli from economic theories with those economists who are seeking effective mathematical tools for their research. A lot of economic problems can be formulated as constrained optimizations and equilibration of their solutions. Various mathematical theories have been supplying economists with indispensable machineries for these problems arising in economic theory. Conversely, mathematicians have been stimulated by various mathematical difficulties raised by economic theories.

The relaxation method has enjoyed an intensive development during many decades and this new edition of this comprehensive text reflects in particular the main achievements in the past 20 years. Moreover, many further improvements and extensions are included, both in the direction of optimal control and optimal design as well as in numerics and applications in materials science, along with an updated treatment of the abstract parts of the theory.

This book giving an exposition of the foundations of modern measure theory offers three levels of presentation: a standard university graduate course, an advanced study containing some complements to the basic course, and, finally, more specialized topics partly covered by more than 850 exercises with detailed hints and references. Bibliographical comments and an extensive bibliography with 2000 works covering more than a century are provided.

A lot of economic problems can formulated as constrained optimizations and equilibration of their solutions.Various mathematical theories have been supplying economists with indispensable machineries for these problems arising in economic theory. Conversely, mathematicians have been stimulated by various mathematical difficulties raised by economic theories. The series is designed to bring together those mathematicians who were seriously interested in getting new challenging stimuli from economic theories with those economists who are seeking for effective mathematical tools for their researchers.