This book focuses on control problems for conservation laws, i.e., equations of the type:

$$\displaystyle \begin{aligned} \partial_t\, u + \partial_x\, f(u) = 0 \qquad u_t+(f(u))_x=0, \end{aligned} $$
(1.1)

where \(u:\mathbb {R}^+\times \mathbb {R} \to \mathbb {R}^n\) is the vector of conserved quantities and \(f:\mathbb {R}^n\to \mathbb {R}^n\) is the flux. Most results will be given for the scalar case (n = 1), but we will present few results valid in the general case.

We consider four types of control problems and use ω to denote the control variables. Namely:

  • Boundary control: We restrict (1.1) to x ∈ [a, b] and control the boundary values: u(t, a) = ω a(t), u(t, b) = ω b(t).

  • Decentralized control: We consider (1.1) on a network and control distribution parameters at nodes.

  • Distributed control: We assume to control some parameters defining the flux, thus f = f(u, ω).

  • Lagrangian control: We assume the flux depends on the position of N controlled particles, f = f(u, y), y = (y 1, …, y N) and \(\dot y_i=\omega _i\).

The interest in conservation laws and their control is motivated by a large and diverse collection of applications. While classical fluid dynamic problems (also covering different areas) motivated research since more than hundred years, a new set of applications motivated recent interests. Many of the latter include problems formulated on networks, which are represented by topological graphs [142]. Among many, let us mention the following: vehicular traffic, water canals, supply chains, air traffic control, data networks, and service networks (gas, water, electricity, etc.).

We give particular attention to vehicular traffic modeling. Classical control problems in this domain correspond to traffic regulation at fixed locations (such as traffic lights, traffic signals, pay tolls, etc.), while the advent of autonomy and communication has opened the possibility of more distributed and ubiquitous controls. More precisely, boundary controls correspond to entrance points and tolls and decentralized controls to traffic signals at junctions. On the other side, variable speed limit gives rise to distributed control and use of autonomy and communication to Lagrangian control problems. Summarizing, vehicular traffic presents potential applications for controls in all the four categories mentioned above. It is well known that conservation laws are strictly connected to Hamilton-Jacobi equations, thus we include a chapter on control of the latter. Also in this case, vehicular traffic applications were among the strongest motivation for researchers.

Many books have been devoted to conservation laws and to their control. Let us mentions the following by categories:

General theory of conservation laws: [51, 99, 213, 236, 251, 252];

Control problems for hyperbolic equations: [32, 103];

Hyperbolic equations on networks: [103, 233];

Hamilton-Jacobi equations: [27, 126, 225];

Modeling of vehicular traffic: [139, 142, 230].

The book can be used for a one semester course at graduate or advanced undergraduate level. The undergraduate students should have been previously exposed to Partial Differential Equations. However, since most of the materials are based on conservation laws, we included Appendix A dealing with the general theory of initial-boundary values problems for balance laws (i.e., including possible of source terms). Readers which are not expert in conservation laws may also want to use as references the textbooks: [51, 99, 251, 252]; On the other side, the theory of conservation laws on networks, i.e., topological graphs, was more recently developed, thus we included Appendix B illustrating the main concepts. The latter are then thoroughly investigated in Chap. 3 to deal with control problems. There are at least three possible course design:

  1. 1.

    Traffic modeling using control of conservation laws. This course would be for investigators more interested in the applications to traffic. Course material: Chap. 1, Appendix B, Chap. 2 (Sects. 2.1, 2.3.2, 2.3.4, 2.4), Chap. 3 (Sects. 3.1, 3.3, 3.4, 3.6), Chap. 4 (Sects. 4.2–4.4), Chap. 5.

  2. 2.

    Mathematical control theory for balance laws. This is for researchers more interested in the mathematical aspects of control. Course material: Chap. 1, Appendix A, Chap. 2 (Sects. 2.1–2.3), Chap. 3 (Sects. 3.1–3.2), Chap. 4 (Sect. 4.1), Chap. 5 (Sects. 5.1–5.3), Chap. 6.

  3. 3.

    Conservation laws on networks and control problems. This is for researchers interested in the general theory of conservation laws on networks and their application. Course material: Chap. 1, Appendix A, Appendix B, Chaps. 2, 3 (Sects. 3.1–3.5), Chap. 4 (Sects. 4.1–4.2), Chap. 6.

The book is organized as follows.

Chapter 2 deals with boundary control problems. We first briefly summarize results for the case of solutions with no shocks (Sect. 2.2). Then illustrate general results for attainable sets (Sect. 2.3), Lyapunov techniques for the scalar case with two boundaries (Sect. 2.4), finally mixed PDE-ODE systems (Sect. 2.5).

Chapter 3 is focused on decentralized controls. These controls act, for instance, at nodes of a network by regulating fluxes. A perfect example is that of traffic lights, ramp metering, and pay tolls. The control problem is formulated in terms of the Riemann solver at nodes (Sect. 3.2), then focusing on signalized junctions (Sect. 3.3), freeway control (Sect. 3.4), and inflow control (Sect. 3.5). We also consider the optimization of travel times and emergency management on networks (Sect. 3.6).

Chapter 4 considers distributed control for conservation laws. The stability of Riemann solvers is a key ingredient to deal with control problems (Sect. 4.1). Our main application is variable speed limit, with results on general control problems (Sect. 4.2), discrete optimization (Sect. 4.3), and systems of equations (Sect. 4.4).

Chapter 5 illustrates Lagrangian control problems. First we introduce coupled ODE-PDE models for moving bottlenecks (Sect. 5.2) and then we develop numerical methods (Sect. 5.3). Applications to traffic management are illustrated both numerically and experimentally (Sect. 5.4).

Chapter 6 explores relations between conservation laws, Hamilton-Jacobi equations, and their control. First strong solutions are considered (Sect. 6.2), then generalized ones (Sect. 6.3). General optimization problems are then discussed (Sect. 6.4).

Appendix A provides a brief introduction to initial and boundary value problems for conservation and balance laws, while Appendix B focuses on conservation laws on networks to model traffic.

The authors wish to thank J.-M. Coron for encouragement during the whole writing process. They are also thankful to Amaury Hayat for contributing to the second chapter and to Alexander Keimer and Christian Claudel for contributing to the sixth chapter.

This work was supported by the following funding: National Science Foundation Cyber-Physical Systems Synergy Grant No. CNS-1837481 and the endowment of the Joseph and Loretta Lopez Chair (B.P.); the Liao-Cho Chair (A.B.). The authors are also thankful to their families for the continuing support and patience during the working of the book.