Exponential stability of general 1-D quasilinear systems with source terms for the C 1 norm under boundary conditions

Amaury Hayat (1, 2)

1 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
2 CaGE - Control And GEometry

We address the question of the exponential stability for the C 1 norm of general 1-D quasilinear systems with source terms under boundary conditions. To reach this aim, we introduce the notion of basic C 1 Lyapunov functions, a generic kind of exponentially decreasing function whose existence ensures the exponential stability of the system for the C 1 norm. We show that the existence of a basic C 1 Lyapunov function is subject to two conditions: an interior condition, intrinsic to the system, and a condition on the boundary controls. We give explicit sufficient interior and boundary conditions such that the system is exponentially stable for the C 1 norm and we show that the interior condition is also necessary to the existence of a basic C 1 Lyapunov function. Finally, we show that the results conducted in this article are also true under the same conditions for the exponential stability in the C p norm, for any p ≥ 1.

Mots clés: Lyapunov; Inhomogeneous; Nonlinear; Hyperbolic systems

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

Amaury Hayat. Exponential stability of general 1-D quasilinear systems with source terms for the C 1 norm under boundary conditions. SIAM Journal on Control and Optimization, 2019, ⟨10.1137/17M1150803⟩. ⟨hal-01613139v3⟩