Optimization in Function Spaces

This book’s title is not very illuminating, but the book is about optimization problems where the variable is a function rather than a scalar or vector. It works on problems of minimizing integrals that depend on an function that we will pick, either (1) with endpoint constraints (the traditional calculus of variations, with the Euler–Lagrange equation), or (2) satisfying a given differential equation, with initial value or boundary value constraints (the beginning of optimal control, with Pontryagin’s principle and Bellman’s equation).

The book starts at the beginning, reminding us how minima are found in single-variable calculus, and then gradually generalizing the ideas of single-variable calculus to multi-variable calculus and then to normed function spaces. The prerequisites are just calculus and a little linear algebra (mostly matrices and eigenvalues), and it doesn’t go very deep into function spaces. It develops all the needed differential equations material. Although it is a “proofs” book, it has an engineering orientation and focuses on solution methods.

The book is very weak on applications. It has numerous examples and exercises, but nearly all merely give the functions and equations and ask for the optimum value. There are complete solutions to all problems in the back of the book.

Another book, that I have not seen but seems comparable, is Mesterton-Gibbons’s A Primer on the Calculus of Variations and Optimal Control Theory.

Bottom line: easy to read with lots of worked examples, and has lots of useful information, but does not go very far.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is allenstenger.com. His mathematical interests are number theory and classical analysis.