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2017 | Book

Mathematical Background Read chapter Read first chapter

A Variational Approach to Nonsmooth Dynamics

Applications in Unilateral Mechanics and Electronics

Author: Prof. Samir Adly

Publisher: Springer International Publishing

Book Series : SpringerBriefs in Mathematics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This brief examines mathematical models in nonsmooth mechanics and nonregular electrical circuits, including evolution variational inequalities, complementarity systems, differential inclusions, second-order dynamics, Lur'e systems and Moreau's sweeping process.

The field of nonsmooth dynamics is of great interest to mathematicians, mechanicians, automatic controllers and engineers. The present volume acknowledges this transversality and provides a multidisciplinary view as it outlines fundamental results in nonsmooth dynamics and explains how to use them to study various problems in engineering. In particular, the author explores the question of how to redefine the notion of dynamical systems in light of modern variational and nonsmooth analysis.

With the aim of bridging between the communities of applied mathematicians, engineers and researchers in control theory and nonlinear systems, this brief outlines both relevant mathematical proofs and models in unilateral mechanics and electronics.

Table of Contents

Frontmatter
Chapter 1. Mathematical Background
Abstract
The main purpose of this chapter is to provide the reader with the knowledge of some basic concepts in convex analysis, nonsmooth analysis and Lyapunov stability theory. Useful mathematical results throughout the book are discussed (without proofs) and illustrated with figures.
Samir Adly
Chapter 2. Nonsmooth Dynamics: An Overview
Abstract
In this chapter, an overview of some mathematical models in nonsmooth dynamics is given. The main purpose is to give the reader a quick but comprehensive snapshot of other classes of nonsmooth systems that can/cannot be captured by the models studied in detail in this book. The following are reviewed: the piecewise dynamical systems; the Filippov concept of solutions for discontinuous differential equations; the notion of differential inclusions along with some general existence results; linear and nonlinear complementarity systems; evolution variational inequalities and their connection with projected dynamical systems; and the so-called measure differential inclusions.
Samir Adly
Chapter 3. Stability Analysis of First-Order Nonsmooth Dynamics
Abstract
The aim of this chapter is to study the existence and uniqueness of solution to a first-order nonsmooth dynamical system involving the subdifferential of a convex, lower semicontinuous and proper function. These problems are also known as evolution variational inequalities. Some conditions ensuring the stability, the asymptotic stability and the finite-time stability of this general class of unilateral dynamics are given. LaSalle’s invariance principle is also developed for this class of problem.
Samir Adly
Chapter 4. Stability Analysis of Second-Order Nonsmooth Dynamics
Abstract
This chapter provides a mathematical theory applicable to the study of second-order dynamic systems with unilateral contact and friction. Conditions ensuring stability (in the sense of Lyapunov), attractivity, and asmptotic stability are given. Some illustrative small-sized examples in unilateral mechanics and in nonregular electrical circuits are also presented. Finally, a rigorous mathematical stability analysis of a DC-DC Buck converter is presented by using tools from nonsmooth and variational analysis.
Samir Adly
Chapter 5. Nonsmooth Lurie Dynamical Systems
Abstract
This chapter is dedicated to the study of Lur’e systems involving maximal monotone and nonmonotone set-valued nonlinearities. The first case is studied with a nonzero feedthrough (or feedforward) matrix D under the so-called passivity condition. In the second case, the matrix D=0 and the problem are formulated into a first-order differential inclusion form where the set-valued right-hand side is upper semicontinuous with nonempty, convex, compact values to obtain the existence of a solution. Then, local hypomonotonicity is assumed to ensure the uniqueness result. A stability analysis and LaSalle’s invariance principle are developed. Finally, some illustrative examples in power electronics are presented.
Samir Adly
Chapter 6. Moreau’s Sweeping Processes
Abstract
This chapter focuses on Moreau’s sweeping processes. Existence and uniqueness results are given when the moving set of constraints is assumed to be convex and absolutely continuous or has a bounded retraction. A new variant of Moreau’s sweeping process with velocity constraint in the moving set is also analyzed. Some applications of the sweeping process to a planning procedure economical model and to the modeling of nonregular electrical circuits are presented.
Samir Adly
Backmatter
Metadata
Title
A Variational Approach to Nonsmooth Dynamics
Author
Prof. Samir Adly
Copyright Year
2017
Electronic ISBN
978-3-319-68658-5
Print ISBN
978-3-319-68657-8
DOI
https://doi.org/10.1007/978-3-319-68658-5

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