jump compatibility relation: $$\begin{aligned}
\label{RH_jump_condition}
f(u_R)-f(u_L)=\sigma (u_R-u_L),\end{aligned}$$ in which case is called a *shock* solution.

Moreover, the solution will be entropic for $\eta$ (according to ) if and only if, $$\begin{aligned}
\label{entropic_shock_condition_system}
q(u_R)-q(u_L)\leq \sigma (\eta(u_R)-\eta(u_L)).\end{aligned}$$ In this case, $(u_L,u_R,\sigma)$ is an *entropic Rankine–Hugoniot discontinuity*.

For a fixed $u_L$, we consider the set of $u_R$ which satisfy and for some $\sigma$. For a general $n\times n$ strictly hyperbolic system of conservation laws endowed with a strictly convex entropy , we know that locally this set of $u_R$ values is made up of $n$ curves (see for example [@lefloch_book p. 140-6]).

The present paper concerns the finite-time stability of piecewise-smooth solutions to , working in the $L^2$ setting. We work in a very general setting. Our techniques are based on the theory of shifts as developed by Vasseur within the context of the relative entropy method (see [@VASSEUR2008323]). We consider systems of the form , with minimal assumptions on the shock families. We ask that the extremal shock speeds (1-shock and n-shock speeds) are separated from the intermediate shock families. If we want to consider 1-shocks, we ask that the 1-shock family satisfy the Liu entropy condition (shock speed decreases as the 