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