No-regret and low-regret controls of space-time fractional parabolic Sturm-Liouville equations in a star graph

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.