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We analyse dissipative boundary conditions for nonlinear hyperbolic systemsin one space dimension. We show that a previous known sufficient condition forexponential stability with respect to the C^1-norm is optimal. In particular aknown weaker sufficient condition for exponential stability with respect to theH^2-norm is not sufficient for the exponential stability with respect to theC^1-norm. Hence, due to the nonlinearity, even in the case of classicalsolutions, the exponential stability depends strongly on the norm considered.We also give a new sufficient condition for the exponential stability withrespect to the W^{2,p}-norm. The methods used are inspired from the theory ofthe linear time-delay systems and incorporate the characteristic method.

Dissipative boundary conditions for nonlinear 1-D hyperbolic systems: sharp conditions through an approach via time-delay systems