For roughly the last thirty-five years, practitioners of control theory have been aware that complex exponentials play a significant role in the solution of control problems governed by partial differential equations. The resulting "non-harmonic" analysis techniques have been described in a variety of journal articles, but

*Fourier Series in Control Theory* represents the first appearance of a synthesis of the relevant literature as well as a collection of new results.

The story begins with the solution of a problem with a vibrating string having two free endpoints. If one can observe only oscillations of the left endpoint of the string of length l over a finite time interval T, what if anything can be said about the unknown initial data? A fairly direct approach using Fourier series and Parseval's equality establishes bounds on the L^{2} norm of the speed of the left endpoint as long as T is at least twice l. However, even a slight modification to the assumptions makes the method unworkable.

The book proceeds to generalize and expand the approach from this simpler problem to solve various controllability problems for vibrating strings, beams, membranes, plates, shells or systems of such. The authors' methods have applications broader than control theory: they offer, for example, a new proof of Bernstein's generalization of Pólya's theorem on singularities of the Dirichlet series.

The prerequisites for the reader are pretty formidable. Knowledge of the basics of linear partial differential equations and familiarity with Lebesgue and Sobelev spaces is assumed, as is a fair piece of introductory functional analysis. From the control theory side, the concepts of controllability, observability and stabilizability are reviewed in the context of systems governed by partial differential equations. Experience with these concepts in the simpler setting of ordinary differential equations would be useful background.

This is primarily a monograph aimed at experts and researchers in the field, although it would be accessible to a graduate student willing to fill in the background. The exposition is clear but terse.

Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.