Optimal bilinear control of stochastic nonlinear Schrödinger equations: mass-(sub)critical case.

We study optimal control problems for stochastic nonlinear Schrödinger equations in both the mass subcritical and critical case. For general initial data of the minimal L 2 regularity, we prove the existence and first order Lagrange condition of an open loop control. In particular, these results apply to the stochastic nonlinear Schrödinger equations with the critical quintic and cubic nonlinearities in dimensions one and two, respectively. Furthermore, we obtain uniform estimates of (backward) stochastic solutions in new spaces of type U 2 and V 2 , adapted to evolution operators related to linear Schrödinger equations with lower order perturbations. These estimates yield a new temporal regularity of (backward) stochastic solutions, which is crucial for the tightness of approximating controls induced by Ekeland's variational principle. [ABSTRACT FROM AUTHOR]