  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 $p\geq1$.
