Optimal Control Problems with Nonlocal Conservation Laws: Existence of Optimizers and Singular Limits in Approximations of Local Conservation Laws

This contribution considers optimal control problems subject to nonlocal conservation laws – those in which the velocity depends nonlocally (i.e., via a convolution) on the solution – and the so-called singular limit. First, the existence of minimizers is demonstrated for a broad class of optimal control problems, involving optimization over the initial datum, velocity, and nonlocal kernel for classical tracking-type $L^2$ cost functionals. Then, it is proven that the obtained minimizers converge to minimizers of the corresponding local optimal control problem when the kernel function of the convolution is of exponential type and approaches a Dirac distribution. Finally, some numerical results are presented.