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## Über dieses Buch

This monograph explores the modeling of conservation and balance laws of one-dimensional hyperbolic systems using partial differential equations. It presents typical examples of hyperbolic systems for a wide range of physical engineering applications, allowing readers to understand the concepts in whichever setting is most familiar to them. With these examples, it also illustrates how control boundary conditions may be defined for the most commonly used control devices.
The authors begin with the simple case of systems of two linear conservation laws and then consider the stability of systems under more general boundary conditions that may be differential, nonlinear, or switching. They then extend their discussion to the case of nonlinear conservation laws and demonstrate the use of Lyapunov functions in this type of analysis. Systems of balance laws are considered next, starting with the linear variety before they move on to more general cases of nonlinear ones. They go on to show how the problem of boundary stabilization of systems of two balance laws by both full-state and dynamic output feedback in observer-controller form is solved by using a “backstepping” method, in which the gains of the feedback laws are solutions of an associated system of linear hyperbolic PDEs. The final chapter presents a case study on the control of navigable rivers to emphasize the main technological features that may occur in real live applications of boundary feedback control.
Stability and Boundary Stabilization of 1-D Hyperbolic Systems will be of interest to graduate students and researchers in applied mathematics and control engineering. The wide range of applications it discusses will help it to have as broad an appeal within these groups as possible.

## Inhaltsverzeichnis

### Chapter 1. Hyperbolic Systems of Balance Laws

In this chapter we provide an introduction to the modeling of balance laws by hyperbolic partial differential equations (PDEs). A balance law is the mathematical expression of the physical principle that the variation of the amount of some extensive quantity over a bounded domain is balanced by its flux through the boundaries of the domain and its production/consumption inside the domain. Balance laws are therefore used to represent the fundamental dynamics of many physical open conservative systems.
Georges Bastin, Jean-Michel Coron

### Chapter 2. Systems of Two Linear Conservation Laws

In this chapter, we start the analysis of the stability and the boundary stabilization design with the simple case of systems of two linear conservation laws. There are two good reasons for beginning in this way.
Georges Bastin, Jean-Michel Coron

### Chapter 3. Systems of Linear Conservation Laws

This chapter mainly deals with the stability of general systems of linear conservation laws under static linear boundary conditions. Depending on whether the issue is examined in the time or in the frequency domain, different stability criteria emerge and are compared, namely from the viewpoint of robustness against uncertainties in the characteristic velocities. The chapter ends with the study of the stability of linear conservation laws under more general boundary conditions that may be dynamic, nonlinear, or switching.
Georges Bastin, Jean-Michel Coron

### Chapter 4. Systems of Nonlinear Conservation Laws

The purpose of this chapter is to extend the exponential stability analysis to the case of systems of nonlinear conservation laws of the form
Georges Bastin, Jean-Michel Coron

### Chapter 5. Systems of Linear Balance Laws

The three previous chapters have dealt with the stability and the boundary stabilization of systems of conservation laws. From now on, we shall move to the analysis of systems of balance laws.
Georges Bastin, Jean-Michel Coron

### Chapter 6. Quasi-Linear Hyperbolic Systems

In this chapter, we continue to explore the use of Lyapunov functions for the stability analysis of quasi-linear hyperbolic systems under dissipative boundary conditions. We address the most general case of systems that cannot be transformed into Riemann coordinates.
Georges Bastin, Jean-Michel Coron

### Chapter 7. Backstepping Control

In this chapter, we address the problem of boundary stabilization of hyperbolic systems of balance laws by full state feedback and by dynamic output feedback in observer-controller form . We consider only the case of systems of two balance laws as in Section 5.3 The control design problem is solved by using a ‘backstepping’ method where the gains of the feedback laws are solutions of an associated system of linear hyperbolic PDEs. The backstepping method for hyperbolic PDEs was initially introduced by Krstic and Smyshlyaev (2008a), Krstic and Smyshlyaev (2008b), and Smyshlyaev et al. (2010). This chapter is essentially based on Vazquez et al. (2011) and Coron et al. (2013).
Georges Bastin, Jean-Michel Coron

### Chapter 8. Case Study: Control of Navigable Rivers

The objective of this chapter is to emphasize the main technological features that may occur in real live applications of boundary feedback control of hyperbolic systems of balance laws. The issue is presented through the specific case study of the control of navigable rivers with a particular focus on the Meuse river in Wallonia (south of Belgium).
Georges Bastin, Jean-Michel Coron

### Backmatter

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