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Table of contents
1 Introduction
1.1 A few general notions
1.2 From controllability to stabilization
1.3 Internal controllability of systems of semilinear coupled one-dimensional wave equations in 1-D with a single control
1.4 Stabilization of hyperbolic systems with a distributed scalar input
1.5 Conclusion and prospects
I Internal controllability of coupled wave equations
2 Internal Controllability of Systems of Semilinear Coupled One-Dimensional Wave Equations with One Control
2.1 Main results and outline of proof
2.2 First case: the linearised system is controllable
2.3 Second case: an example with an uncontrollable linearized system
2.4 Further questions
II Stabilization of 1-D linear hyperbolic systems with a scalar input
3 Internal rapid stabilization of a 1-D linear transport equation with a scalar feedback
3.1 Introduction
3.2 Denition and properties of the transformation
3.3 Well-posedness and stability of the closed-loop system
3.4 Further remarks and questions
4 Finite-time internal stabilization of a linear 1-D transport equation
4.1 Introduction
4.2 The exponentially stable semigroup
4.3 The limit semigroup
4.4 An explicit example
4.5 Comments and further questions
5 Exponential stabilization of the linearized water tank system
5.1 Introduction
5.2 Properties of the system and presentation of the method
5.3 Dealing with mass conservation
5.4 Controllability
5.5 A heuristic construction
5.6 Backstepping transformation and feedback law
Appendix 5.A Proof of Proposition 5.2.1
Appendix 5.B Proof of Proposition 5.6.1
Appendix 5.C Expression of the feedback coecients before and after variable changes
