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

Consider a scalar conservation law with discontinuous flux (1): \begin{equation*} \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} \quad \quad \quad(1) \end{equation*} where u = u(x, t) is the state variable and fl, fr 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)≐StABu-(x) denote the solution of the Cauchy problem for (1), with initial datum u(⋅,0)=u-, that satisfy at x = 0 the interface entropy condition associated to a connection (A, B) (see Adimurthi, S. Mishra and G.D. Veerappa Gowda, J. Hyperbolic Differ. Equ. 2 (2005) 783–837), we analyze the family of profiles that can be attained by (1) at a given time T > 0: \mathcal{A}^{AB}(T)=\left\{\mathcal{S}_T^{AB} \,\overline u : \ \overline u\in{\bf L}^\infty(\mathbb{R})\right\}.\ We provide a full characterization of AAB(T) as a class of functions in BVloc(ℝ\{0}) that satisfy suitable Oleǐnik-type inequalities, and that admit one-sided limits at x = 0 which satisfy specific conditions related to the interface entropy criterion. Relying on this characterisation, we establish the Lloc1-compactness of the set of attainable profiles when the initial data u- vary in a given class of uniformly bounded functions, taking values in closed convex sets. We also discuss some applicationsof these results to optimization problems arising in traffic flow.