  Consider a scalar conservation law with discontinuous flux
\begin{equation*}\tag{1} \quad u_{t}+f(x,u)_{x}=0, \qquad f(x,u)= \begin{cases}
f_l(u)\ &\text{if}\ x<0,\\ f_r(u)\ & \text{if} \ x>0, \end{cases}
\end{equation*} where $u=u(x,t)$ is the state variable and $f_{l}$, $f_{r}$ are
strictly convex maps. We study the Cauchy problem for (1) from the point of
view of control theory regarding the initial datum as a control. Letting
$u(x,t)\doteq \mathcal{S}_t^{AB} \overline u(x)$ denote the solution of the
Cauchy problem for (1), with initial datum $u(\cdot,0)=\overline u$, that
satisfy at $x=0$ the interface entropy condition associated to a connection
$(A,B)$ (see~\cite{MR2195983}), we analyze the family of profiles that can be
attained by (1) at a given time $T>0$: \begin{equation*}
\mathcal{A}^{AB}(T)=\left\{\mathcal{S}_T^{AB} \,\overline u : \ \overline
u\in{\bf L}^\infty(\mathbb{R})\right\}. \end{equation*} We provide a full
characterization of $\mathcal{A}^{AB}(T)$ as a class of functions in
$BV_{loc}(\mathbb{R}\setminus\{0\})$ that satisfy suitable Ole\v{\i}nik-type
inequalities, and that admit one-sided limits at $x=0$ which satisfy specific
conditions related to the interface entropy criterium. Relying on this
characterisation, we establish the ${\bf L^1}_{loc}$-compactness of the set of
attainable profiles when the initial data $\overline u$ vary in a given class
of uniformly bounded functions, taking values in closed convex sets. We also
discuss some applications of these results to optimization problems arising in
porous media flow models for oil recovery and in traffic flow.

Analysis of PDEs;Optimization and Control