Abstract

We consider optimal control problems subject to nonlocal conservation laws. Nonlocal conservation laws are conservation laws where the velocity of the corresponding dynamics depends on an integral term of the solution, in contrast to local conservation laws where the dependency is on the solution at a given space-time point. Nonlocal conservation laws behave mathematically quite differently, weak solutions are unique and an entropy condition is thus not required.

We will give a short overview of the currently existing results concerning existence and uniqueness of solution and incorporate also results on the singular limit problem (do solutions converge towards the local entropy solution when the integral kernel converges to a Dirac distribution?).

We will then investigate optimal control problems for nonlocal conservation laws and present conditions on the functions to optimize over (velocity field, nonlocal kernel, initial datum) for which the optimal control problem admits a solution. We conclude with results regarding the singular limit of optimal control problems subject to nonlocal dynamics, i.e., whether solutions of the nonlocal optimal control problem converge towards solutions of the corresponding local optimal control problem when the nonlocal kernel converges towards a Dirac distribution. Some numerical illustrations will conclude the talk.