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We address the question of the exponential stability for the $C^{1}$ norm ofgeneral 1-D quasilinear systems with source terms under boundary conditions. Toreach this aim, we introduce the notion of basic $C^{1}$ Lyapunov functions, ageneric kind of exponentially decreasing function whose existence ensures theexponential stability of the system for the $C^{1}$ norm. We show that theexistence of a basic $C^{1}$ Lyapunov function is subject to two conditions: aninterior condition, intrinsic to the system, and a condition on the boundarycontrols. We give explicit sufficient interior and boundary conditions suchthat the system is exponentially stable for the $C^{1}$ norm and we show thatthe interior condition is also necessary to the existence of a basic $C^{1}$Lyapunov function. Finally, we show that the results conducted in this articleare also true under the same conditions for the exponential stability in the$C^{p}$ norm, for any $p\geq1$.

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