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