1R9. Mathematical Control Theory of Coupled PDEs. - I Lasiecka (Univ of Virginia, Charlottesville VA). SIAM, Philadelphia. 2002. 242 pp. Softcover. ISBN 0-89871-486-9. $60.00.
Reviewed by GC Gaunaurd (Code AMSRL-SE-RU, Army Res Lab, 2800 Powder Mill Rd, Adelphi MD 20783-1197).
The mathematics of control theory for a single Partial Differential Equation has been studied for some time. However, for more complex systems governed by systems of coupled PDEs, the available works are fewer. The tools developed for single PDE systems are usually inadequate for the analysis of coupled systems. New questions have been formulated, including how can one take advantage of the coupling in the model to improve the system performance? The propagation of some components of a system into some other, originate new phenomena via the coupling.
The present book describes classes of coupled PDE models displaying the above-mentioned coupled properties and presents tools to analyze the resulting control problems. The ultimate goal of the book is said to be to provide a mathematical theory to guide the solution of three main problems: a) well-posedness and regularity, b) stabilization and stability, and c) optimal control and existence and uniqueness of some associated Riccati equations.
The structural acoustics model is used toward the end of the book as the choice “example” to illustrate various coupling phenomena that appear in interconnected systems. There are wave equations in an acoustic medium that appear coupled to the plate or shell equations from which they are separated by an interface that is part of the boundary for the acoustic medium.
The book has six chapters. It is only possible here to give their titles. The analysis starts with the well-posedness of 2nd-order nonlinear equations with boundary damping. It continues with a study of the stabilizability of nonlinear waves and plates. There is then a chapter on the uniform stability of structural acoustic models and another on Semi Group and PDE models for structural acoustic control problems. The final two chapters deal with feedback noise control for finite and infinite time-horizon problems. This results in detailed studies of certain pertinent Riccati equations.
Mathematical Control Theory of Coupled PDEs has 242 pages, with no figures and no computed results. This is a pure mathematical treatment full of theorems, lemmas, assumptions, propositions, and countless corollaries. It is the author’s stated hope that it will be of use to applied acousticians and theoretical engineers, but this is an unrealistic expectation. The connection to structural acoustics is buried in a sea of theorems with little applicability. There are over 200 references, but about half are by the author herself and one of her associates. The couple of works mentioned which are authored by well-known acousticians, such as C Fuller and the textbook by P Morse and U Ingard, are the only ones that come from the regular acoustic literature. Therefore, this mathematical document will be mostly of interest to other mathematicians carrying on research on these obscure topics. This is certainly not a textbook, but perhaps could be a reference mathematics book for some institutional libraries. With the current emphasis on relevance required by the Army Research Office (ARO), now a part of the Army Research Lab, this reviewer was surprised to learn that they sponsored this work.