Stability and Convergence of Mechanical Systems with Unilateral Constraints

Preface

Contents

Notation

1 Introduction

1.1 Motivation

1.2 Historical Notes on the Theory of Stability

1.3 Non-smooth Dynamical Systems

1.4 Stability and Convergence

1.5 Literature Survey

1.6 Objective and Scope

1.7 Outline

2 Non-smooth Analysis

2.1 Sets

2.2 Functions and Continuity

2.3 Generalised Derivatives

2.4 Set-valued Functions

2.5 De.nitions from Convex Analysis

2.6 Subderivative

2.7 Summary

3 Measure and Integration Theory

3.1 Measures

3.2 The Lebesgue Integral

3.3 Signed Measures

3.4 Real Measures

3.5 Di.erential Measures

3.6 The Differential Measure of a Bilinear Form

3.7 Summary

4 Non-smooth Dynamical Systems

4.1 Differential Equations

4.2 Differential Inclusions

4.3 Measure Differential Inclusions

4.4 Summary

5 Mechanical Systems with Set-valued Force Laws

5.1 Non-smooth Potential Theory

5.2 Contact Geometry

5.3 Force Laws for Frictional Unilateral Contact

5.4 Measure Newton-Euler Equations

5.5 Summary

6 Lyapunov Stability Theory for Measure Differential Inclusions

6.1 Mathematical Preliminaries

6.2 Invariant Sets and Limit Sets

6.3 Definitions of Stability Properties for Autonomous Systems

6.4 Definitions of Stability Properties of Solutions Non-autonomous Systems

6.5 Basic Lyapunov Theorems of Autonomous Systems

6.6 LaSalle’s Invariance Principle

6.7 Instability

6.8 Summary

7 Stability Properties in Mechanical Systems

7.1 Total Mechanical Energy

7.2 Stability Results for Mechanical Systems

7.3 Attractivity of Equilibrium Sets

7.4 Instability of Equilibrium Positions and Sets

7.5 Examples

7.6 Summary

8 Convergence Properties of Monotone Measure Di.erential Inclusions

8.1 Convergent Systems

8.2 Convergence of Maximal Monotone Systems

8.3 Tracking Control for Lur’e Type Systems

8.4 Illustrative Examples

8.5 Summary

9 Concluding Remarks

Sources and Translations

References

Index