Exponential boundary feedback stabilization of a shock steady state for the inviscid Burgers equation

Georges Bastin (1) , Jean-Michel Coron (2, 3) , Amaury Hayat (4, 3) , Peipei Shang (5, 4, 6)

1 CSAM - Centre for Systems Engineering and Applied Mechanics
2 Département de Mathématiques-Université de Paris XI
3 CaGE - Control And GEometry
4 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
5 School of Mathematical Sciences [Shanghai]
6 D-MATH - Department of Mathematics [ETH Zurich]

In this paper, we study the exponential stabilization of a shock steady state for the inviscid Burgers equation on a bounded interval. Our analysis relies on the construction of an explicit strict control Lyapunov function. We prove that by appropriately choosing the feedback boundary conditions, we can stabilize the state as well as the shock location to the desired steady state in H 2-norm, with an arbitrary decay rate.

Domaines: Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]

Georges Bastin, Jean-Michel Coron, Amaury Hayat, Peipei Shang. Exponential boundary feedback stabilization of a shock steady state for the inviscid Burgers equation. 2018. ⟨hal-01723361v1⟩