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Focusing on the foundational aspects of the Calculus of Variations, this book is designed for students with no prior background, making it self-contained. It begins with essential concepts in Functional Analysis and Sobolev spaces, progressing to one-dimensional variational problems with numerous examples and exercises. The text also covers continuous and compact operators, leading to multi-dimensional Sobolev spaces and complex problems. With a didactic approach, it aims to build a solid understanding, encouraging students to explore more advanced topics in the field.

Focusing on optimization, this resource introduces science and engineering students to mathematical programming, calculus of variations, and optimal control. It emphasizes key concepts and the significance of optimality conditions within these areas. The book also offers systematic presentation of affordable approximation methods and includes exercises of varying difficulty to enhance the learning experience.

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.

This undergraduate textbook introduces students of science and engineering to the fascinating field of optimization. It is a unique book that brings together the subfields of mathematical programming, variational calculus, and optimal control, thus giving students an overall view of all aspects of optimization in a single reference. As a primer on optimization, its main goal is to provide a succinct and accessible introduction to linear programming, nonlinear programming, numerical optimization algorithms, variational problems, dynamic programming, and optimal control. Prerequisites have been kept to a minimum, although a basic knowledge of calculus, linear algebra, and differential equations is assumed.

Weak convergence is essential in modern nonlinear analysis due to its compactness properties similar to finite-dimensional spaces, where bounded sequences correspond to weakly relatively compact sets. However, weak convergence presents challenges with nonlinear functionals and operations, complicating nonlinear analysis beyond expectations. Parametrized measures help to understand weak convergence and its interaction with nonlinear functionals, providing a means to represent weak limits of compositions with nonlinear functions through integrals. This approach is particularly useful for analyzing oscillatory phenomena and tracking changes in oscillations when nonlinear functionals are applied. Additionally, weak convergence is crucial in the calculus of variations, as uniform bounds in norm for sequences enable the existence of weakly convergent subsequences. Establishing weak lower semicontinuity is vital for demonstrating the existence of minimizers for specific functionals, representing a key step in the direct method of the calculus of variations. Significant research has focused on identifying conditions under which weak lower semicontinuity holds for nonlinear functionals expressed as integrals. The findings indicate that some form of convexity, interpreted broadly, is typically involved in these scenarios.