## Stability and Boundary Stabilization of 1-D Hyperbolic SystemsThis 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. |

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### Contents

1 | |

2 Systems of Two Linear Conservation Laws | 55 |

3 Systems of Linear Conservation Laws | 85 |

4 Systems of Nonlinear Conservation Laws | 117 |

5 Systems of Linear Balance Laws | 159 |

6 QuasiLinear Hyperbolic Systems | 202 |

7 Backstepping Control | 219 |

Control of Navigable Rivers | 229 |

B WellPosedness of the Cauchy Problem for QuasiLinear Hyperbolic Systems | 255 |

C Properties and Comparisons of the Functions ρ ρ2 and ρ | 260 |

D Proof of Lemma 412 b and c | 281 |

E Proof of Theorem 511 | 285 |

F Notations | 293 |

295 | |

305 | |

A WellPosedness of the Cauchy Problem for Linear Hyperbolic Systems | 243 |

### Other editions - View all

Stability and Boundary Stabilization of 1-D Hyperbolic Systems Georges Bastin,Jean-Michel Coron No preview available - 2016 |

Stability and Boundary Stabilization of 1-D Hyperbolic Systems Georges Bastin,Jean-Michel Coron No preview available - 2018 |

### Common terms and phrases

ÃÂ assume balance laws Bastin boundary conditions boundary control boundary feedback Boundary Stabilization Cauchy problem Chapter characteristic equation characteristic form characteristic velocities class C1 closed-loop coefficients compatibility conditions constant control law control system Coron defined definition denotes density Differential Equations dissipative boundary conditions dynamics eigenvalues Euler equations exists exponential stability exponentially converge feedback control flow rate gates H-co hyperbolic systems inequalities initial condition input knCi L2-norm linear conservation laws linear systems Lyapunov function Lyapunov stability matrix method of characteristics Meuse river nonlinear notations open channel PI control poles positive proof of Theorem Proposition QL.t quadratic quadratic form quasi-linear hyperbolic systems Riemann coordinates Saint-Venant equations satisfying the compatibility scalar Section set-point solution stability analysis stability condition steady systems of balance systems of conservation Theorem well-posedness