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# Calculus of variations

See below for a selection of the latest books from Calculus of variations category. Presented with a red border are the Calculus of variations books that have been lovingly read and reviewed by the experts at Lovereading. With expert reading recommendations made by people with a passion for books and some unique features Lovereading will help you find great Calculus of variations books and those from many more genres to read that will keep you inspired and entertained. And it's all free!

## Variational Methods For Evolving Objects

This volume consists of eight original survey papers written by invited lecturers in connection with a conference 'Variational Methods for Evolving Objects' held at Hokkaido University, Sapporo, Japan, July 30 - August 3, 2012. The topics of papers vary widely from problems in image processing to dynamics of topological defects, and all involve some nonlinear phenomena of current major research interests. These papers are carefully prepared so that they serve as a good starting point of investigation for graduate students and new comers to the field and are strongly recommended.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets except North America

## Calculus of Variations and Partial Differential Equations of First Order

Format: Hardback Release Date: 27/11/2020

In this second English edition of Caratheodory's famous work (originally published in German), the two volumes of the first edition have been combined into one (with a combination of the two indexes into a single index). There is a deep and fundamental relationship between the differential equations that occur in the Calculus of Variations and partial differential equations of the first order: in particular, to each such partial differential equation there correspond variational problems. This basic fact forms the rationale for Caratheodory's masterpiece. Includes a Guide to the Literature and an Index.

## Uniform Rectifiability and Quasiminimizing Sets of Arbitrary Codimension

Format: Paperback / softback Release Date: 27/11/2020

Roughly speaking, a $d$-dimensional subset of $\mathbfR^n$ is minimizing if arbitrary deformations of it (in a suitable class) cannot decrease its $d$-dimensional volume. For quasiminimizing sets, one allows the mass to decrease, but only in a controlled manner. To make this precise we follow Almgren's notion of 'restricted sets' [{\textbold 2}]. Graphs of Lipschitz mappings $f\:\mathbfR^d \to \mathbfR^{n-d}$ are always quasiminimizing, and Almgren showed that quasiminimizing sets are rectifiable. Here we establish uniform rectifiability properties of quasiminimizing sets, which provide a more quantitative sense in which these sets behave like Lipschitz graphs. (Almgren also established stronger smoothness properties under tighter quasiminimality conditions.)Quasiminimizing sets can arise as minima of functionals with highly irregular 'coefficients'. For such functionals, one cannot hope in general to have much more in the way of smoothness or structure than uniform rectifiability, for reasons of bilipschitz invariance. (See also [{\textbold 9}].) One motivation for considering minimizers of functionals with irregular coefficients comes from the following type of question. Suppose that one is given a compact set $K$ with upper bounds on its $d$-dimensional Hausdorff measure, and lower bounds on its $d$-dimensional topology.What can one say about the structure of $K$? To what extent does it behave like a nice $d$-dimensional surface? A basic strategy for dealing with this issue is to first replace $K$ by a set which is minimizing for a measurement of volume that imposes a large penalty on points which lie outside of $K$. This leads to a kind of regularization of $K$, in which cusps and very scattered parts of $K$ are removed, but without adding more than a small amount from the complement of $K$. The results for quasiminimizing sets then lead to uniform rectifiability properties of this regularization of $K$. To actually produce minimizers of general functionals it is sometimes convenient to work with (finite) discrete models. A nice feature of uniform rectifiability is that it provides a way to have bounds that cooperate robustly with discrete approximations, and which survive in the limit as the discretization becomes finer and finer.

## New Directions in Dirichlet Forms

Format: Hardback Release Date: 27/11/2020

The theory of Dirichlet forms brings together methods and insights from the calculus of variations, stochastic analysis, partial differential and difference equations, potential theory, Riemannian geometry and more. This book features contributions by leading experts and provides up-to-date, authoritative accounts on exciting developments in the field and on new research perspectives.Topics covered include the following: stochastic analysis on configuration spaces, specifically a mathematically rigorous approach to the stochastic dynamics of Gibbs measures and infinite interacting particle systems; subelliptic PDE, homogenization, and fractals; geometric aspects of Dirichlet forms on metric spaces and function theory on such spaces; generalized harmonic maps as nonlinear analogues of Dirichlet forms, with an emphasis on non-locally compact situations; and a stochastic approach based on Brownian motion to harmonic maps and their regularity. Various new connections between the topics are featured, and it is demonstrated that the theory of Dirichlet forms provides the proper framework for exploring these connections.

## Variational and Optimal Control Problems on Unbounded Domains

This volume contains the proceedings of the workshop on Variational and Optimal Control Problems on Unbounded Domains, held in memory of Arie Leizarowitz, from January 9-12, 2012, in Haifa, Israel. The workshop brought together a select group of worldwide experts in optimal control theory and the calculus of variations, working on problems on unbounded domains. The papers in this volume cover many different areas of optimal control and its applications. Topics include needle variations in infinite-horizon optimal control, Lyapunov stability with some extensions, small noise large time asymptotics for the normalized Feynman-Kac semigroup, linear-quadratic optimal control problems with state delays, time-optimal control of wafer stage positioning, second order optimality conditions in optimal control, state and time transformations of infinite horizon problems, turnpike properties of dynamic zero-sum games, and an infinite-horizon variational problem on an infinite strip. This book is co-published with Bar-Ilan University (Ramat-Gan, Israel).

## Lectures on the Calculus of Variations

Format: Hardback Release Date: 27/11/2020

Based on lectures delivered at the AMS meeting in 1901, this book describes the progress in calculus of variations made in the last 30 years of the nineteenth century. Among other topics, the author describes the landmark results of Weierstrass on sufficient conditions for the extremum of a functional in terms of the second variation. Also discussed are Kneser's sufficient conditions, Weierstrass' theory of the isoperimetric problem, and Hilbert's theorem on the existence of an extremum of an integral. Although the original book was written nearly 100 years ago, it remains very useful in learning about classical calculus of variations.

## Computational Intelligence and Optimization Methods for Control Engineering

This volume presents some recent and principal developments related to computational intelligence and optimization methods in control. Theoretical aspects and practical applications of control engineering are covered by 14 self-contained contributions. Additional gems include the discussion of future directions and research perspectives designed to add to the reader's understanding of both the challenges faced in control engineering and the insights into the developing of new techniques. With the knowledge obtained, readers are encouraged to determine the appropriate control method for specific applications.

## An Introduction to Linear and Nonlinear Scattering Theory

This monograph has two main purposes, first to act as a companion volume to more advanced texts by gathering together the principal mathematical topics commonly used in developing scattering theories and, in so doing, provide a reasonable, self-contained introduction to linear and nonlinear scattering theory for those who might wish to begin working in the area. Secondly, to indicate how these various aspects might be applied to problems in mathematical physics and the applied sciences. Of particular interest will be the influence of boundary conditions.

## Turnpike Conditions in Infinite Dimensional Optimal Control

This book provides a comprehensive study of turnpike phenomenon arising in optimal control theory. The focus is on individual (non-generic) turnpike results which are both mathematically significant and have numerous applications in engineering and economic theory. All results obtained in the book are new. New approaches, techniques, and methods are rigorously presented and utilize research from finite-dimensional variational problems and discrete-time optimal control problems to find the necessary conditions for the turnpike phenomenon in infinite dimensional spaces. The semigroup approach is employed in the discussion as well as PDE descriptions of continuous-time dynamics. The main results on sufficient and necessary conditions for the turnpike property are completely proved and the numerous illustrative examples support the material for the broad spectrum of experts. Mathematicians interested in the calculus of variations, optimal control and in applied functional analysis will find this book a useful guide to the turnpike phenomenon in infinite dimensional spaces. Experts in economic and engineering modeling as well as graduate students will also benefit from the developed techniques and obtained results.

## Nonlinear Combinatorial Optimization

Graduate students and researchers in applied mathematics, optimization, engineering, computer science, and management science will find this book a useful reference which provides an introduction to applications and fundamental theories in nonlinear combinatorial optimization. Nonlinear combinatorial optimization is a new research area within combinatorial optimization and includes numerous applications to technological developments, such as wireless communication, cloud computing, data science, and social networks. Theoretical developments including discrete Newton methods, primal-dual methods with convex relaxation, submodular optimization, discrete DC program, along with several applications are discussed and explored in this book through articles by leading experts.

## Stochastic Linear-Quadratic Optimal Control Theory: Open-Loop and Closed-Loop Solutions

This book gathers the most essential results, including recent ones, on linear-quadratic optimal control problems, which represent an important aspect of stochastic control. It presents the results in the context of finite and infinite horizon problems, and discusses a number of new and interesting issues. Further, it precisely identifies, for the first time, the interconnections between three well-known, relevant issues - the existence of optimal controls, solvability of the optimality system, and solvability of the associated Riccati equation. Although the content is largely self-contained, readers should have a basic grasp of linear algebra, functional analysis and stochastic ordinary differential equations. The book is mainly intended for senior undergraduate and graduate students majoring in applied mathematics who are interested in stochastic control theory. However, it will also appeal to researchers in other related areas, such as engineering, management, finance/economics and the social sciences.