Title: Stabilization of 1D hyperbolic systems

Abstract: Hyperbolic systems in one space dimension are omnipresent in nature. For this reason, the stabilization of their steady-states is an interesting problem. We will discuss this question in the case of a boundary or a pointwise control. We will first look at "density-velocity systems", a class that encompasses many physical equations: isentropic Euler equations, Saint-Venant equations, osmosis model, etc. We show that this class of equations has a remarkable property: it can be stabilized by simple boundary feedback laws for any size of domain. We will also see how to generalize these results respectively to shock steady-states and Proportional-Integral (PI) controllers. Shocks are an important phenomena in hyperbolic systems, while PI controllers are widely used in practice, but still quite hard to handle mathematically for nonlinear infinite dimensional systems. Finally, we will talk about stabilizing second order systems modelling traffic flows with a pointwise control. This control can be, for instance, an autonomous vehicle. Traffic flows interacting with autonomous vehicles are very interesting from a control perspective: the relevant solutions can be non-entropic, raising new questions and a need for new tools.