##
**Nonsmooth analysis and control theory.**
*(English)*
Zbl 1047.49500

Graduate Texts in Mathematics 178. New York, NY: Springer (ISBN 0-387-98336-8/hbk). xii, 276 p. (1998).

From the Preface: “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 back 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.

One reason for the growth of the subject has been, without a doubt, the recognition that nondifferentiable phenomena are more widespread, and play a more important role, than had been thought. Philosophically at least, this is in keeping with the coming to the fore of several other types of irregular and nonlinear behavior: catastrophes, fractals, and chaos.

In recent years, nonsmooth analysis has come to play a role in functional analysis, optimization, optimal design, mechanics and plasticity, differential equations (as in the theory of viscosity solutions), control theory, and, increasingly, in analysis generally (critical point theory, inequalities, fixed point theory, variational methods…). In the long run, we expect its methods and basic constructs to be viewed as a natural part of differential analysis.

We have found that it would be relatively easy to write a very long book on nonsmooth analysis and its applications; several times, we did. We have now managed not to do so, and in fact our principal claim for this work is that it presents the essentials of the subject clearly and succinctly, together with some of its applications and a generous supply of interesting exercises. We have also incorporated in the text a number of new results which clarify the relationships between the different schools of thought in the subject. We hope that this will help 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.

We now proceed to discuss the contents. Chapter 0 is an Introduction in which we allow ourselves a certain amount of hand-waving. The intent is to give the reader an avant-goût of what is to come, and to indicate at an early stage why the subject is of interest.

There are many exercises in Chapters 1 to 4, and we recommend (to the active reader) that they be done. Our experience in teaching this material has had a great influence on the writing of this book, and indicates that comprehension is proportional to the exercises done. The end-of-chapter problems also offer scope for deeper understanding. We feel no guilt in calling upon the results of exercises later as needed.

Chapter 1, on proximal analysis, should be done carefully by every reader of this book. We have chosen to work here in a Hilbert space, although the greater generality of certain Banach spaces having smooth norms would be another suitable context. We believe the Hilbert space setting makes for a more accessible theory on first exposure, while being quite adequate for later applications.

Chapter 2 is devoted to the theory of generalized gradients, which constitutes the other main approach (other than proximal) to developing nonsmooth analysis. The natural habitat of this theory is Banach space, which is the choice made. The relationship between these two principal approaches is now well understood, and is clearly delineated here. As for the preceding chapter, the treatment is not encyclopedic, but covers the important ideas.

In Chapter 3 we develop certain special topics, the first of which is value function analysis for constrained optimization. This topic is previewed in Chapter 0, and §3.1 is helpful, though not essential, in understanding certain proofs in the latter part of Chapter 4. The next topic, mean value inequalities, offers a glimpse of more advanced calculus. It also serves as a basis for the solvability results of the next section, which features the Graves-Lyusternik Theorem and the Lipschitz Inverse Function Theorem. Section 3.4 is a brief look at a third route to nonsmooth calculus, one that bases itself upon directional subderivates. It is shown that the salient points of this theory can be derived from the earlier results. We also present here a self-contained proof of Rademacher’s Theorem. In §3.5 we develop some machinery that is used in the following chapter, notably measurable selection. We take a quick look at variational functionals, but by-and-large, the calculus of variations has been omitted. The final section of the chapter examines in more detail some questions related to tangency.

Chapter 4 (“A short course in control theory”), as its title implies, is a self-contained introduction to the theory of control of ordinary differential equations. This is a biased introduction, since one of its avowed goals is to demonstrate virtually all of the preceding theory in action. It makes no attempt to address issues of modeling or of implementation. Nonetheless, most of the central issues in control are studied, and we believe that any serious student of mathematical control theory will find it essential to have a grasp of the tools that are developed here via nonsmooth analysis: invariance, viability, trajectory monotonicity, viscosity solutions, discontinuous feedback, and Hamiltonian inclusions. We believe that the unified and geometrically motivated approach presented here for the first time has merits that will continue to make themselves felt in the subject…”

One reason for the growth of the subject has been, without a doubt, the recognition that nondifferentiable phenomena are more widespread, and play a more important role, than had been thought. Philosophically at least, this is in keeping with the coming to the fore of several other types of irregular and nonlinear behavior: catastrophes, fractals, and chaos.

In recent years, nonsmooth analysis has come to play a role in functional analysis, optimization, optimal design, mechanics and plasticity, differential equations (as in the theory of viscosity solutions), control theory, and, increasingly, in analysis generally (critical point theory, inequalities, fixed point theory, variational methods…). In the long run, we expect its methods and basic constructs to be viewed as a natural part of differential analysis.

We have found that it would be relatively easy to write a very long book on nonsmooth analysis and its applications; several times, we did. We have now managed not to do so, and in fact our principal claim for this work is that it presents the essentials of the subject clearly and succinctly, together with some of its applications and a generous supply of interesting exercises. We have also incorporated in the text a number of new results which clarify the relationships between the different schools of thought in the subject. We hope that this will help 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.

We now proceed to discuss the contents. Chapter 0 is an Introduction in which we allow ourselves a certain amount of hand-waving. The intent is to give the reader an avant-goût of what is to come, and to indicate at an early stage why the subject is of interest.

There are many exercises in Chapters 1 to 4, and we recommend (to the active reader) that they be done. Our experience in teaching this material has had a great influence on the writing of this book, and indicates that comprehension is proportional to the exercises done. The end-of-chapter problems also offer scope for deeper understanding. We feel no guilt in calling upon the results of exercises later as needed.

Chapter 1, on proximal analysis, should be done carefully by every reader of this book. We have chosen to work here in a Hilbert space, although the greater generality of certain Banach spaces having smooth norms would be another suitable context. We believe the Hilbert space setting makes for a more accessible theory on first exposure, while being quite adequate for later applications.

Chapter 2 is devoted to the theory of generalized gradients, which constitutes the other main approach (other than proximal) to developing nonsmooth analysis. The natural habitat of this theory is Banach space, which is the choice made. The relationship between these two principal approaches is now well understood, and is clearly delineated here. As for the preceding chapter, the treatment is not encyclopedic, but covers the important ideas.

In Chapter 3 we develop certain special topics, the first of which is value function analysis for constrained optimization. This topic is previewed in Chapter 0, and §3.1 is helpful, though not essential, in understanding certain proofs in the latter part of Chapter 4. The next topic, mean value inequalities, offers a glimpse of more advanced calculus. It also serves as a basis for the solvability results of the next section, which features the Graves-Lyusternik Theorem and the Lipschitz Inverse Function Theorem. Section 3.4 is a brief look at a third route to nonsmooth calculus, one that bases itself upon directional subderivates. It is shown that the salient points of this theory can be derived from the earlier results. We also present here a self-contained proof of Rademacher’s Theorem. In §3.5 we develop some machinery that is used in the following chapter, notably measurable selection. We take a quick look at variational functionals, but by-and-large, the calculus of variations has been omitted. The final section of the chapter examines in more detail some questions related to tangency.

Chapter 4 (“A short course in control theory”), as its title implies, is a self-contained introduction to the theory of control of ordinary differential equations. This is a biased introduction, since one of its avowed goals is to demonstrate virtually all of the preceding theory in action. It makes no attempt to address issues of modeling or of implementation. Nonetheless, most of the central issues in control are studied, and we believe that any serious student of mathematical control theory will find it essential to have a grasp of the tools that are developed here via nonsmooth analysis: invariance, viability, trajectory monotonicity, viscosity solutions, discontinuous feedback, and Hamiltonian inclusions. We believe that the unified and geometrically motivated approach presented here for the first time has merits that will continue to make themselves felt in the subject…”

### MSC:

49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |

49J52 | Nonsmooth analysis |

58C20 | Differentiation theory (Gateaux, Fréchet, etc.) on manifolds |

93-02 | Research exposition (monographs, survey articles) pertaining to systems and control theory |