  We develop a version of Haar and Holmgren methods which applies to
discontinuous solutions of nonlinear hyperbolic systems and allows us to
control the L1 distance between two entropy solutions. The main difficulty is
to cope with linear hyperbolic systems with discontinuous coefficients. Our
main observation is that, while entropy solutions contain compressive shocks
only, the averaged matrix associated with two such solutions has compressive or
undercompressive shocks, but no rarefaction-shocks -- which are recognized as a
source for non-uniqueness and instability. Our Haar-Holmgren-type method rests
on the geometry associated with the averaged matrix and takes into account
adjoint problems and wave cancellations along generalized characteristics. It
extends the method proposed earlier by LeFloch et al. for genuinely nonlinear
systems. In the present paper, we cover solutions with small total variation
and a class of systems with general flux that need not be genuinely nonlinear
and includes for instance fluid dynamics equations. We prove that solutions
generated by Glimm or front tracking schemes depend continuously in the L1 norm
upon their initial data, by exhibiting an L1 functional controling the distance
between two solutions.
