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In the last decades the subject of nonsmooth analysis has grown rapidly due to the recognition that nondifferentiable phenomena are more widespread, and play a more important role, than had been thought. In recent years, it has come to play a role in functional analysis, optimization, optimal design, mechanics and plasticity, differential equations, control theory, and, increasingly, in analysis. This volume presents the essentials of the subject clearly and succinctly, together with some of its applications and a generous supply of interesting exercises. The book begins with an introductory chapter which gives the reader a sampling of what is to come while indicating at an early stage why the subject is of interest. The next three chapters constitute a course in nonsmooth analysis and identify a coherent and comprehensive approach to the subject leading to an efficient, natural, yet powerful body of theory. The last chapter, as its name implies, is a self-contained introduction to the theory of control of ordinary differential equations. End-of-chapter problems also offer scope for deeper understanding. The authors have incorporated in the text a number of new results which clarify the relationships between the different schools of thought in the subject. Their goal is to make nonsmooth analysis accessible to a wider audience. In this spirit, the book is written so as to be used by anyone who has taken a course in functional analysis.



0. Introduction

Among the issues that routinely arise in mathematical analysis are the following three:
  • to minimize a function f(x);
  • to solve an equation F(x) = y for x as a function of y; and
  • •|to derive the stability of an equilibrium point x* of a differential equation x = φ(x).

1. Proximal Calculus in Hilbert Space

We introduce in this chapter two basic constructs of nonsmooth analysis: proximal normals (to a set) and proximal subgradients (of a function). Proximal normals are direction vectors pointing outward from a set, generated by projecting a point onto the set. Proximal subgradients have a certain local support property to the epigraph of a function. It is a familiar device to view a function as a set (through its graph), but we develop the duality between functions and sets to a much greater extent, extending it to include the calculus of these normals and subgradients. The very existence of a proximal subgradient often says something of interest about a function at a point; the Density Theorem of §3 is a deep result affirming existence on a substantial set. From it we deduce two minimization principles. These are theorems bearing upon situations where a minimum is “almost attained,” and which assert that a small perturbation leads to actual attainment. We will meet some useful classes of functions along the way: convex, Lipschitz, indicator, and distance functions. Finally, we will see some elements of proximal calculus, notably the sum and chain rules.

2. Generalized Gradients in Banach Space

The calculus of generalized gradients is the best-known and most frequently invoked part of nonsmooth analysis. Unlike proximal calculus, it can be developed in an arbitrary Banach space X. In this chapter we make a fresh start in such a setting, but this time, in contrast to Chapter 1, we begin with functions and not sets. We present the basic results for the class of locally Lipschitz functions. Then the associated geometric concepts are introduced, including for the first time a look at tangency. In fact, we examine two notions of tangency; sets for which they coincide are termed regular and enjoy useful properties. We proceed to relate the generalized gradient to the constructs of the preceding chapter when X is a Hilbert space. Finally, we derive a useful limiting-gradient characterization when the underlying space is finite dimensional.

Special Topics

In this chapter we study a number of different issues, each of interest in its own right. All the results obtained here build upon, and in some cases complement, those of the preceding chapters, and several of them address problems discussed in the Introduction. This is the case, for example, of the first section on constrained optimization. Some of the results of §§5 and 6 will be called upon in Chapter 4.

4. A Short Course in Control Theory

Mathematics, as well as several areas of application, abounds with situations where it is desired to control the behavior of the trajectories of a given dynamical system. The goal can be either geometric (keep the state of the system in a given set, or bring it toward the set), or functional (find the trajectory that is optimal relative to a given criterion). More specific issues arise subsequently, such as the construction of feedback control mechanisms achieving the aims we have in mind. In this chapter we will identify a complex of such fundamental and related issues, as they arise in connection with the control of ordinary differential equations in a deterministic setting. The first section sets the scene and develops a technical base for the entire chapter.


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