On boundary stability of inhomogeneous 2 × 2 1-D hyperbolic systems for the C¹ norm

Sorbonne Université, Université Paris-Diderot SPC, CNRS, INRIA, Laboratoire Jacques-Louis Lions, LJLL, équipe CAGE, 75005 Paris, France.

^* Corresponding author: hayat@ljll.math.upmc.fr

Received: 11 May 2018
Accepted: 31 October 2018

Abstract

We study the exponential stability for the C¹ norm of general 2 × 2 1-D quasilinear hyperbolic systems with source terms and boundary controls. When the propagation speeds of the system have the same sign, any nonuniform steady-state can be stabilized using boundary feedbacks that only depend on measurements at the boundaries and we give explicit conditions on the gain of the feedback. In other cases, we exhibit a simple numerical criterion for the existence of basic C¹ Lyapunov function, a natural candidate for a Lyapunov function to ensure exponential stability for the C¹ norm. We show that, under a simple condition on the source term, the existence of a basic C¹ (or C^p, for any p ≥ 1) Lyapunov function is equivalent to the existence of a basic H² (or H^q, for any q ≥ 2) Lyapunov function, its analogue for the H² norm. Finally, we apply these results to the nonlinear Saint-Venant equations. We show in particular that in the subcritical regime, when the slope is larger than the friction, the system can always be stabilized in the C¹ norm using static boundary feedbacks depending only on measurements at the boundaries, which has a large practical interest in hydraulic and engineering applications.