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

Abstract

We study optimal control problems for stochastic nonlinear Schrodinger 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 Schrodinger 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 Schrodinger 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.