A Variational Approach to Nonsmooth Dynamics

1. Front Matter

   Pages i-xv

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2. Mathematical Background

   • Samir Adly

   Pages 1-37

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3. Nonsmooth Dynamics: An Overview

   • Samir Adly

   Pages 39-51

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4. Stability Analysis of First-Order Nonsmooth Dynamics

   • Samir Adly

   Pages 53-69

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5. Stability Analysis of Second-Order Nonsmooth Dynamics

   • Samir Adly

   Pages 71-102

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6. Nonsmooth Lurie Dynamical Systems

   • Samir Adly

   Pages 103-126

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7. Moreau’s Sweeping Processes

   • Samir Adly

   Pages 127-148

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8. Back Matter

   Pages 149-159

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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.

• nonsmooth dynamics
• Lyapunov stability
• stability analysis nonsmooth dynamics
• Moreau sweeping process
• nonsmooth mechanics
• evolution variational inequalities
• complementarity systems
• differential inclusions
• invariance principle
• unilateral dynamical systems
• nonregular electrical circuits
• DC-DC Buck converters
• Lur'e systems
• 34A60, 34H15, 37N40, 46N10, 47J22, 49J52
• 49J53, 65K10, 65K15, 90C33

“The book under review treats the subject of nonsmooth dynamical systems using tools from convex and variational analysis. … This book is well written, pedagogic and self-contained. It is accessible to a large audience, including mathematicians, engineers and undergraduate students with a good knowledge of ODEs.” (Daniel Goeleven, Mathematical Reviews, September, 2018)​

• Laboratoire XLIM, Université de Limoges, Limoges, France

  Samir Adly

Samir Adly received a master's degree in Mathematical Engineering (1992) and a Ph.D. in Applied Mathematics (1995) from the University of Limoges, France, where he is now full professor of mathematics. He is also director of the mathematics and security information department of the XLIM research institute (UMR CNRS) and head of the MOD team (Modeling, Optimization, Dynamics) at XLIM. Adly's research interests focus on theoretical and algorithmic issues related to variational inequalities, complementarity problems, optimization, nonsmooth dynamical systems, variational analysis and its applications. He has published more than 70 articles in peer-reviewed journals and has been working in the area of nonsmooth dynamical systems since 2002.

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