We analyze a class of control problems where the initial datum acts as a control and the state is given by the entropy solution of (local) conservation laws by a nonlocal-to-local limiting strategy. In particular we characterize the limit up to subsequence of minimizers to nonlocal control problems as minimizer of the corresponding local ones. Moreover, we also prove an analogous result at a discrete level by means of a Eulerian-Lagrangian scheme.
Friedrich et al. (Wed,) studied this question.