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A Primer on the Calculus of Variations and Optimal Control Theory

Mike Mesterton-Gibbons
American Mathematical Society
Publication Date: 
Number of Pages: 
Student Mathematical Library 50
[Reviewed by
John P. Curran
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This book achieves exactly what it sets out to do: It gives a thorough introduction to the topics given in its title, with minimal prerequisites. (The only prerequisite is some knowledge of differential equations.) From a mathematical point of view, it is a good book, especially useful for undergraduates and beginning graduate students. Why, then, does the book seem so dull?

I think that the author is a victim of his own conscientiousness. The level of detail in the book is unusual, because you can follow the details of each argument without resorting to pencil and paper. Every nuance of notation is spelled out. Every possibility is supported by examples. Exercises are carefully footnoted. These are all admirable in a textbook, but their cumulative effect drags down the reader. The size of the book also contributes to this effect. The pages are small (14 cm x 22 cm), and this leads to a lot of a page flipping. Also, there are many displayed formulas, as many as 10 per page, which slows the eye down. All of these formulas are numbered, but not all of them are referred to later on.

To be fair, when the book turns to control theory, Mesterton-Gibbons cuts himself some slack, and the book livens up a bit. He also seems to enjoy himself more, considering meaty examples drawn from applications. Perhaps the book should have started with control theory, history and logical development notwithstanding.

John Curran is Assistant Professor of Mathematics, Eastern Michigan University, where he coordinates the actuarial science program. He worked for a Wall Street firm for several years before obtaining his Ph.D. in applied mathematics from Brown University.