Optimal bilinear control of stochastic nonlinear Schr\"odinger equations: mass-(sub)critical case

We study optimal bilinear control problems for stochastic nonlinear Schr\"odinger equations in both the mass subcritical and critical case. For general initial data of the minimal L2 regularity, we prove the existence and first order Lagrange condition of an open loop control. Furthermore, we obtain uniform estimates of (backward) stochastic solutions in new spaces of type U2 and V2, adapted to evolution operators related to linear Schr\"odinger equations with lower order perturbations. In particular, we obtain a new temporal regularity of rescaled (backward) stochastic solutions, which is the key ingredient in the proof of tightness of approximating controls induced by Ekeland's variational principle.