Nonsmooth analysis refers to differential analysis in the absence of differentiability. It can be regarded as a subfield of that vast subject known as nonlinear analysis. While nonsmooth analysis has classical roots (we claim to have traced its lineage hack to Dini), it is only in the last decades that the subject has grown rapidly. To the point, in fact, that further development has sometimes appeared in danger of being stymied, due to the plethora of definitions and unclearly related theories.

Preface

List of Figures

0 Introduction

1 Analysis Without Lineaxization

2 Flow-Invariant Sets

3 Optimization

4 Control Theory

5 Notation

1 Proximal Calculus in Hilbert Space

1 Closest Points and Proximal Normals

2 Proximal Subgradients

3 The Density Theorem

4 Minimization Principles

5 Quadratic Inf-Convolutions

6 The Distance Function

7 Lipschitz Functions

8 The Sum Rule

9 The Chain Rule

10 Limiting Calculus

11 Problems on Chapter 1

2 Generalized Gradients in Banach Space

1 Definition and Basic Properties

2 Basic Calculus

3 Relation to Derivatives

4 Convex and Regular Functions

5 Tangents and Normals

6 Relationship to Proximal Analysis

7 The Bouligand Tangent Cone and Regular Sets

8 The Gradient Formula in Finite Dimensions

9 Problems on Chapter 2

3 Special Topics

1 Constrained Optimization and Value Functions

2 The Mean Value Inequality

3 Solving Equations

4 Derivate Calculus and Rademacher's Theorem

5 Sets in L2 and Integral b-~nctionals

6 Tangents and Interiors

7 Problems on Chapter 3

4 A Short Course in Control Theory

1 Trajectories of DiffercntiM Inclusions

2 Weak Invariance

3 Lipschitz Dependence and Strong Invariance

4 Equilibria

5 Lyapounov Theory and Stabilization

6 Monotonicity and Attainability

7 The Hamilton Jacobi Equation and Viscosity Solutions

8 Feedback Synthesis from Semisolutions

9 Necessary Conditions for Optimal Control

10 Normality and Controllability

11 Problems on Chapter 4

Notes and Comments

List of Notation

Bibliography

Index