The classical tools of optimal control theory yield the best strategy for going from where you are right now to where you want to be in the future. But what if your target is selected randomly and is only revealed at a random later time T? Should you just do nothing until this happens? Should you only optimize the expected total cost or can you also provide some guarantees about the worst-case scenario?

I will use simple 1- and 2-dimensional examples to show how "free boundaries" and discontinuities arise based on our answers to the above questions. These phenomena pose different computational challenges & influence our choice of discretization/solution strategy for PDEs encoding the optimality.

This talk is meant to be self-contained and will not assume any prior background in control theory, dynamic programming, or Hamilton-Jacobi PDEs.