We consider the relaxation model

u_t + v_x = 0
v_t + u_x = (f(u)-v)/\epsilon

with u in R^n, f: R^n \mapsto R^n. We assume that Df is 
strictly hyperbolic with eigenvalues strictly less 
than 1 and the initial data (u_0,v_0) have small 
total variation.

We prove that the solution (u^\epsilon, v^\epsilon) is well defined for 
all t > 0, and its total variation satisfies a uniform bound, independent 
of t, \epsilon. Moreover, as \epsilon tends to 0, the solutions 
(u^\epsilon, v^\epsilon) converge to a unique limit (u(t), v(t)): u(t) 
is the unique entropic solution of the corresponding hyperbolic system 

u_t + F(u)_x = 0 and v(t,x) = F(u(t,x)) for all t > 0, a.e. x in R.

The proof relies on the introduction of a new functional for 
the solutions of (u^\epsilon,v^\epsilon), corresponding to the 
Glimm-Liu interaction potential for the waves of same families.