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

Consider a scalar conservation law with discontinuous flux (1): 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: We provide a full characterization of AAB(T) as a class of functions in BVloc(0) that satisfy suitable Oleinik-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