Stability of front tracking solutions to the initial and boundary value problem for systems of conservation laws

We deal with the non characteristic initial and boundary value problem for an $n\times n$ strictly hyperbolic system of conservation laws in one space dimension % $$ \partial_t u+ \partial_x F(u)=0,\qquad u(0,x) = \bar u (x)\,,\qquad b\big( u(\psi(t),t) \big) = g(t)\,.\eqno (\ast) $$ % Here $F$ is a smooth vector field defined in an open, convex neighborhood of the origin of $\real^n$, $\bar u$ and $g$ are functions with small total variation, $x=\psi(t)$ is a non characteristic Lipschitz boundary profile, and $b$ a $\mathcal{C}^1$ function. We prove that the front tracking solutions to ($\ast$) constructed by D. Amadori in \cite{Amadori} are stable for the $\elleuno$ topology. This implies the existence of a Standard Riemann Semigroup and hence the well-posedness of ($\ast$).