Boundary stability of 1-D nonlinear inhomogeneous hyperbolic systems for the C¹ norm

We address the question of the exponential stability for the C¹ norm of general one-dimensional quasilinear systems with source terms under boundary conditions. To reach this aim, we introduce the notion of basic C¹ Lyapunov functions, a generic kind of exponentially decreasing function whose existence ensures the exponential stability of the system for the C¹ norm. We show that the existence of a basic C¹ 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 C¹ norm and we show that the interior condition is also necessary to the existence of a basic C¹ Lyapunov function.