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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 $f_{l}$, $f_{r}$ arestrictly convex maps. We study the Cauchy problem for (1) from the point ofview 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 theCauchy problem for (1), with initial datum $u(\cdot,0)=\overline u$, thatsatisfy at $x=0$ the interface entropy condition associated to a connection$(A,B)$ (see~\cite{MR2195983}), we analyze the family of profiles that can beattained by (1) at a given time $T>0$: \begin{equation*}\mathcal{A}^{AB}(T)=\left\{\mathcal{S}_T^{AB} \,\overline u : \ \overlineu\in{\bf L}^\infty(\mathbb{R})\right\}. \end{equation*} We provide a fullcharacterization of $\mathcal{A}^{AB}(T)$ as a class of functions in$BV_{loc}(\mathbb{R}\setminus\{0\})$ that satisfy suitable Ole\v{\i}nik-typeinequalities, and that admit one-sided limits at $x=0$ which satisfy specificconditions related to the interface entropy criterium. Relying on thischaracterisation, we establish the ${\bf L^1}_{loc}$-compactness of the set ofattainable profiles when the initial data $\overline u$ vary in a given classof uniformly bounded functions, taking values in closed convex sets. We alsodiscuss some applications of these results to optimization problems arising inporous media flow models for oil recovery and in traffic flow.

Attainable profiles for conservation laws with flux function spatially discontinuous at a single point