Many real-world optimal control problems can be most concisely modeled as mixed systems of differential and algebraic equations. This formulation allows for problems to be cleanly decomposed into component parts that are connected in a networked fashion. As one might expect, dealing with such systems can be rather complex, from both theoretical and computational standpoints, and this book attempts to provide a compact reference to the latest advances in both theory and software for doing so.
The book is based primarily on material presented at a workshop of the same name, also organized by the editors. As such, each chapter is written by different authors and presented as a self-contained work with its own introduction, body, conclusions, and bibliography. The resulting book is intended for applied mathematicians, engineers, and computational scientists, and assumes the reader is familiar and comfortable with control and optimization theory.
Chapter 1, written by the editors, introduces differential-algebraic equations (DAEs) and how the general optimal control problem can be formulated and solved with them. It also serves to introduce the notation and terminology used throughout the book, as well as to encapsulate and tie together the subsequent chapters. Chapters 2 through 4 discuss the theory behind many practical problems that arise when working with DAEs. Chapter 5 describes the StratiGraph software package for analyzing and visualizing descriptor system models. Chapters 6 through 8 discuss the analysis and control of linear DAEs. Chapters 9 through 12 deal with solutions to optimal control problems formulated as DAEs. Finally, chapters 13 through 16 demonstrate the use of many of the techniques covered in the book on example problems.
The format of the book as a collection of (mostly) independent works lends itself very well to its intended use as a topical reference, but it can also make browsing the book somewhat tedious, as the basic problem descriptions and formulations are repeated in many chapters. That said, a little redundancy can be forgiven, as the editors did a commendable job of ensuring both notational and thematic consistency. And while writing styles do vary noticeably between chapters, this does not detract from readability.
Outside of chapter 1, which is primarily an overview and survey, chapters 2 and 4 are perhaps the most broadly accessible. Chapter 2 gives a treatment of regularization and linearization of DAEs, discussing how to formulate control problems to ensure that computational solution methods do not fail. Chapter 4 gives a treatment for examining asymptotic behavior and growth rate for more complex linear time-varying systems through the use of recent developments in spectral theory.
The final four chapters of the book, which demonstrate how the techniques may be applied to actual problems, are also particularly useful. These examples act as guides for how researchers can use the techniques presented, and for what kind of results they can expect.
Because much of the book is based on workshop material, some of the chapters are rather dense, spending most of their time proving their thesis rather than discussing or interpreting the results. This can make a casual reading of the chapters difficult, and may turn readers off to what they might otherwise find interesting. On the other hand, if the reader happens to be seeking a solution to the specific problem a chapter treats, this may be helpful.
In all, the book provides an interesting overview of some of the recent advances in DAE-based simulation and optimal control, but the reader should be aware that the intended audience should already be familiar and comfortable with the topic. This is not an introductory or learning text. And, while neither sufficient as, nor intended to be, a comprehensive reference, it provides enough information about many important problems to aid further investigation.
