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 $$L^2$$ 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 $$U^2$$ U 2 and $$V^2$$ 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.

中文翻译：

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随机非线性薛定谔方程的最优双线性控制：质量（亚）临界情况

我们研究了质量亚临界和临界情况下随机非线性薛定谔方程的最优控制问题。对于最小$$L^2$$L 2 正则性的一般初始数据，我们证明了开环控制的存在性和一阶拉格朗日条件。特别是，这些结果分别适用于具有一维和二维临界五次非线性和三次非线性的随机非线性薛定谔方程。此外，我们在 $$U^2$$ U 2 和 $$V^2$$ V 2 类型的新空间中获得（向后）随机解的统一估计，适用于与具有低阶扰动的线性薛定谔方程相关的进化算子. 这些估计产生了（向后）随机解的新时间规律，