  We are concerned with a space-time fractional parabolic initial-boundary
value problem of Sturm Liouville type in a general star graph with mixed
Dirichlet and Neumann boundary controls. We first give several existence,
uniqueness and regularity results of weak and very-weak solutions. Using the
notion of no-regret control introduced by Lions, we prove the existence,
uniqueness, and characterize the low regret control of a quadratic boundary
optimal control problem, then we prove that this low regret control converges
to the no-regret control and we provide the associated optimality systems and
conditions that characterize that no-regret control.
