Variational calculus and optimal control. Optimization with elementary convexity.
2nd ed.

*(English)*Zbl 0865.49001
Undergraduate Texts in Mathematics. New York, NY: Springer-Verlag. xv, 461 p. (1996).

This is the second edition of an admirable undergraduate text on the calculus of variations. The first edition, “Variational calculus with elementary convexity” (1983), has been reviewed elsewhere (Zbl 0523.49001; M.R. 84f:49001). The book’s first two parts have been revised, and a third part concerning optimal control has been added (about 80 pages).

The book offers a clear and comprehensive introduction to the classical calculus of variations, in which one is to choose a function \(y\) of the real variable \(x\) in order to minimize an integral expression of the form \(\int^b_af(x,y(x),y'(x))dx\) subject to various endpoint, isoperimetric, or differential constraints. (Some important problems where the unknown function \(y\) depends on a vector variable \(x\) are mentioned, but the main thrust of the book concerns problems where \(x\) is scalar.) The context is designed to be accessible even to readers who have not yet mastered the Lebesgue theory of integration: the first part of the book develops the principal results for minimization over smooth functions \(y\), while the second and third allowing for arcs \(y\) whose derivatives are only piecewise continuous. This does mean that some advanced topics, better suited for graduate-level instruction, do not appear – the existence of minimizers by the direct method, for example, is beyond the scope of this work. It does not mean, however, that the book is soft on theory – on the contrary, careful justification of every assertion is provided, and the presentation is managed to encourage the reader always to keep the applicable hypotheses clearly in view. A glance at the book’s table of contents (below) reveals the scope of its coverage: there is enough material here to support not only a 30-hour introduction to the calculus of variations, but also a subsequent 30-hour sequence treating classical mechanics and optimal control. In addition to its choice of topics and polished writing and presentation, a noteworthy feature of the book is the philosophical position that guides the author’s exposition from first to last. Troutman writes “The goal remains to solve problems completely (and exactly) whenever possible at the mathematical level required to formulate them.” The emphasis on problem-solving comes through clearly in the wealth of solved examples and exercises the book contains. The text itself is full of interesting solved examples, and the substantial problems grouped at the end of each chapter number over three hundred in total. Enough of these are straightforward applications of the theory that students can consolidate their mastery of the conceptual core of the subject, but many more are stimulating excursions into applied problems of real intrinsic interest, drawn from antiquity to the present day. Solving problems on their own level requires rethinking the presentation and emphasis given to standard topics. This is particularly important in optimization problems, where a stationary point need not be a solution: some kind of elementary sufficient condition for optimality is needed. In this book, convexity fills this role. It is well known that any stationary point for a (strictly) convex functional is a (unique) global minimizer; one of the key contributions of this book is the systematic exploitation of this fact in the aid of elementary, self-contained, and complete solutions to many classical variational problems whose convex features are not at all evident at first glance. The brachistrochrone problem, the catenary problem, Dido’s isoperimetric problem, and Zenodoros’ problem of maximal volume enclosed by a surface of revolution with given area, are completely solved by this treatment after suitable transformations (Sections 3.5 and 8.8). Many problems with isoperimetric and Lagrangian constraints are shown to yield to similar methods. A common feature of all these solutions is that they start with the sufficient conditions, reasoning, “If only we could arrange…then we would have a guaranteed minimizer.” This approach to problem-solving is just the opposite of the step-by-step recipes all too common in undergraduate textbooks: it calls for considerable originality and some ad-hoc analysis that is just what students need to develop mathematical maturity. The necessity of such standard conditions as the Euler-Lagrange equations and the Pontryagin minimum principle is discussed only after their power has been convincingly demonstrated in convex cases where they are actually sufficient for optimality. The Lagrange multiplier rule receives a particularly fine treatment. The text’s emphasis on elementary sufficient conditions based on convexity, and its success in many standard problems, makes it possible to treat fields of extremals and the Hamilton-Jacobi equation as an “Advanced Topic”; this important part of the classical approach is given a thorough presentation in Chapter 9. The author describes his subject’s classical roots with care, but also points out its latest developments. This shows not only in the applied problems discussed in examples and exercises, but also dramatically in the extensive bibliography, which includes many recent references. Readers of this book will find not only a meticulous and compelling self-contained presentation, but also a valuable set of directions in which they can set out to pursue further studies.

Contents: 0. Review of optimization in \(\mathbb{R}^d\).

Part One – Basic Theory: 1. Standard optimization problems; 2. Linear spaces and GĂ˘teaux variations; 3. Minimization of convex functions; 4. The lemmas of Lagrange and DuBois-Reymond; 5. Local extrema in normed linear spaces; 6. The Euler-Lagrange equations.

Part Two – Advanced Topics: 7. Piecewise \(C^1\) extremal functions; 8. Variational principles in mechanics; 9. Sufficient conditions for a minimum.

Part Three – Optimal Control: 10. Control problems and sufficiency considerations; 11. Necessary conditions for optimality.

The book offers a clear and comprehensive introduction to the classical calculus of variations, in which one is to choose a function \(y\) of the real variable \(x\) in order to minimize an integral expression of the form \(\int^b_af(x,y(x),y'(x))dx\) subject to various endpoint, isoperimetric, or differential constraints. (Some important problems where the unknown function \(y\) depends on a vector variable \(x\) are mentioned, but the main thrust of the book concerns problems where \(x\) is scalar.) The context is designed to be accessible even to readers who have not yet mastered the Lebesgue theory of integration: the first part of the book develops the principal results for minimization over smooth functions \(y\), while the second and third allowing for arcs \(y\) whose derivatives are only piecewise continuous. This does mean that some advanced topics, better suited for graduate-level instruction, do not appear – the existence of minimizers by the direct method, for example, is beyond the scope of this work. It does not mean, however, that the book is soft on theory – on the contrary, careful justification of every assertion is provided, and the presentation is managed to encourage the reader always to keep the applicable hypotheses clearly in view. A glance at the book’s table of contents (below) reveals the scope of its coverage: there is enough material here to support not only a 30-hour introduction to the calculus of variations, but also a subsequent 30-hour sequence treating classical mechanics and optimal control. In addition to its choice of topics and polished writing and presentation, a noteworthy feature of the book is the philosophical position that guides the author’s exposition from first to last. Troutman writes “The goal remains to solve problems completely (and exactly) whenever possible at the mathematical level required to formulate them.” The emphasis on problem-solving comes through clearly in the wealth of solved examples and exercises the book contains. The text itself is full of interesting solved examples, and the substantial problems grouped at the end of each chapter number over three hundred in total. Enough of these are straightforward applications of the theory that students can consolidate their mastery of the conceptual core of the subject, but many more are stimulating excursions into applied problems of real intrinsic interest, drawn from antiquity to the present day. Solving problems on their own level requires rethinking the presentation and emphasis given to standard topics. This is particularly important in optimization problems, where a stationary point need not be a solution: some kind of elementary sufficient condition for optimality is needed. In this book, convexity fills this role. It is well known that any stationary point for a (strictly) convex functional is a (unique) global minimizer; one of the key contributions of this book is the systematic exploitation of this fact in the aid of elementary, self-contained, and complete solutions to many classical variational problems whose convex features are not at all evident at first glance. The brachistrochrone problem, the catenary problem, Dido’s isoperimetric problem, and Zenodoros’ problem of maximal volume enclosed by a surface of revolution with given area, are completely solved by this treatment after suitable transformations (Sections 3.5 and 8.8). Many problems with isoperimetric and Lagrangian constraints are shown to yield to similar methods. A common feature of all these solutions is that they start with the sufficient conditions, reasoning, “If only we could arrange…then we would have a guaranteed minimizer.” This approach to problem-solving is just the opposite of the step-by-step recipes all too common in undergraduate textbooks: it calls for considerable originality and some ad-hoc analysis that is just what students need to develop mathematical maturity. The necessity of such standard conditions as the Euler-Lagrange equations and the Pontryagin minimum principle is discussed only after their power has been convincingly demonstrated in convex cases where they are actually sufficient for optimality. The Lagrange multiplier rule receives a particularly fine treatment. The text’s emphasis on elementary sufficient conditions based on convexity, and its success in many standard problems, makes it possible to treat fields of extremals and the Hamilton-Jacobi equation as an “Advanced Topic”; this important part of the classical approach is given a thorough presentation in Chapter 9. The author describes his subject’s classical roots with care, but also points out its latest developments. This shows not only in the applied problems discussed in examples and exercises, but also dramatically in the extensive bibliography, which includes many recent references. Readers of this book will find not only a meticulous and compelling self-contained presentation, but also a valuable set of directions in which they can set out to pursue further studies.

Contents: 0. Review of optimization in \(\mathbb{R}^d\).

Part One – Basic Theory: 1. Standard optimization problems; 2. Linear spaces and GĂ˘teaux variations; 3. Minimization of convex functions; 4. The lemmas of Lagrange and DuBois-Reymond; 5. Local extrema in normed linear spaces; 6. The Euler-Lagrange equations.

Part Two – Advanced Topics: 7. Piecewise \(C^1\) extremal functions; 8. Variational principles in mechanics; 9. Sufficient conditions for a minimum.

Part Three – Optimal Control: 10. Control problems and sufficiency considerations; 11. Necessary conditions for optimality.

Reviewer: P.Loewen (Bath)

##### MSC:

49-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to calculus of variations and optimal control |