- Author: Alexandre Bayen, Maria Laura Delle Monache, Mauro Garavello, Paola Goatin, and Benedetto Piccoli
- Series: Progress in Nonlinear Differential Equations and Their Applications
- Publisher: Springer
- Publication Date: 04/24/2022
- Number of Pages: 227
- Format: Hardcover
- Price: $59.99
- ISBN: 978-3-030-93014-1
- Category: monograph
[Reviewed by Bill Satzer, on 07/28/2025]
Control theory develops analyses, creates models, and produces designs that enable dynamical systems to achieve certain desirable behaviors using feedback and control methods. It has a strong theoretical basis in mathematics, and it is widely used in several engineering disciplines. This book is aimed at graduate students. The authors suggest that it is also accessible to upper-level graduates with some background in partial differential equations. However, some acquaintance with control theory would be highly desirable.
Control theory is a broad subject with a considerable variety of specialized areas and applications. In this book, the focus is on applications to traffic control. Traffic should be construed broadly because if can include not only vehicular traffic, but also data networks, supply chains, and service networks like gas, water and electricity. Nonetheless, the emphasis of this book is on vehicular traffic. In this and many related applications the theory is formulated on networks that are represented by topological graphs.
As the title indicates, the book focuses on approaches that use conservation laws for control problems. That means that it looks at equations of the form $\partial_{t}u + \partial_xf(u) = 0$ and $u_t + (f(u))_x = 0$ where $u : \mathbb{R}^{+} \times \mathbb{R} \rightarrow \mathbb{R}^n$ is the vector of conserved quantities and $f : \mathbb{R}^n \rightarrow \mathbb{R}^n$ is the flux. The authors mostly work in the scalar case $(n = 1)$, but provide results for the general case.
$$Conservation laws express the condition that whatever elements are moving in traffic (vehicles, data packets, or volumes of water, for example) are constantly present and never disappear. Flux constraints enable limits to traffic flow at selected points. Four main types of control problems are considered. These are categorized as boundary control, decentralized control, distributed control, and Lagrangian control. For vehicular traffic, boundary controls may be used for entry points, decentralized controls for traffic signals at junctions, distributed control for applying variable speed limits, and Lagrangian control for autonomous vehicles.
Simulations are provided as illustrative examples. One of them models the behavior of vehicles at a large traffic circle in Rome.
While this is intended to be a text, it may also function as a reference for those interested in vehicular traffic applications.
Although the book has several authors (and additional ones identified as contributors to specific chapters), the book maintains a consistent style and approach. The bibliography is very extensive. There are no exercises.
Bill Satzer (bsatzer@gmail.com), now retired from 3M Company, spent most of his career as a mathematician working in industry on a variety of applications. He did his PhD work in dynamical systems and celestial mechanics.