Stabilization of some nonlinear hyperbolic PDEs

Abstract: Abstract: We consider two types of systems : density-velocity systems and traffic flows. Density-velocity systems encompass many physical equations: isentropic Euler equations, Saint-Venant equations, osmosis model, etc. We show that these equations have a local dissipative property that allows to stabilize any steady-state with boundary feedback controls, provided some physical assumption. Moreover, this holds even if we have no knowledge of some the system parameters or with a single control. Traffic flows are very interesting from a control perspective. In many situations the steady-states are unstable, leading to travelling waves, known as stop-and-go waves by engineers or simply jam. From a mathematical point of view they can be represented by coupled hyperbolic PDEs with solutions of class BV. We will present on-going work showing how one can try to stabilize the steady-states using autonomous vehicles, i.e. pointwise controls. This leads to a system of ODEs and PDEs coupled by a flux relation, which provoke non-classical shocks. The solutions are then at most BV and the control is contained in the dynamics of the ODEs.

analysis of PDEsoptimization and control

Audience: learners

Control in Times of Crisis. Online seminar.

Series comments: The link of each talk will be sent to the mailing list of the seminar. In order to be included in the list, please send a message to one of the organizers.