In this talk, we will review several methods based on a Lyapunov approach to stabilize quasilinear hyperbolic systems with proportional controls at the boundaries. We will start with a generic method for the $H^{2}$ and $C^{1}$ norm before focusing on fluid equations. In particular, we will see that the general Saint-Venant system, a well-known model for shallow waters used in practice for the regulation of navigable rivers, has a particular structure that enables the tabilization of any of its regular steady-states by simple boundary controls, whatever the source term is, even if the physical data associated (slope, friction, etc.) are unknown. This feat comes from the existence of a remarkable local entropy that we will discuss. We will also see how to stabilize a shock steady-state for the Burgers' equation and the Saint-Venant equations. Finally, we will discuss the limit of these proportional boundary controls and talk about the design of more sophisticated controls: Proportional-Integral controllers which are very much used in practice while remaining quite hard to handle mathematically for infinite dimensional systems.

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