Structure of solutions and initial data identification for conservation laws with spatial discontinuous flux

In this thesis we deal with two control problems for a scalar conservation law with spatial discontinuous flux. The first result concerns the characterization of the attainable set at time T>0 in terms of some Oleinik-type inequalities and some further conditions related to the fact that some profiles might be reachable only by solutions containing a shock. The second part of the thesis deals with the problem of initial data identification. We generalize the known results for a scalar conservation law with convex flux to our discontinuous flux setting, adapting the method of generalized characteristics by Dafermos. In particular we prove that the set of initial data that lead to the same profile at time T is either a singleton or an infinite dimensional cone that, unlike the convex flux case, is not always convex.