  We show uniqueness and stability in $L^2$ and for all time for
piecewise-smooth solutions to hyperbolic balance laws. We have in mind
applications to gas dynamics, the isentropic Euler system and the full Euler
system for a polytropic gas in particular. We assume the discontinuity in the
piecewise smooth solution is an extremal shock. We use only mild hypotheses on
the system. Our techniques and result hold without smallness assumptions on the
solutions. We can handle shocks of any size. We work in the class of bounded,
measurable solutions satisfying a single entropy condition. We also assume a
strong trace condition on the solutions, but this is weaker than
$BV_{\text{loc}}$. We use the theory of a-contraction (see Kang and Vasseur
[Arch. Ration. Mech. Anal., 222(1):343--391, 2016]) developed for the stability
of pure shocks in the case without source.
