The primary objective of this paper is to directly establish the observability inequality for stochastic parabolic equations from measurable sets. In an immediate practical application, our focus centers on the investigation of optimal actuator placement to achieve minimum norm controls in the context of approximative controllability for stochastic parabolic equations. We introduce a comprehensive formulation of the optimization problem, encompassing both the determination of the actuator location and the corresponding minimum norm control. More precisely, we reformulate the problem into a two-player zero-sum game scenario, resulting in the derivation of four equivalent formulations. Ultimately, we substantiate the pivotal outcome that the solution to the relaxed optimization problem serves as the optimal actuator placement for the classical problem.

本文的主要目标是从可测集直接建立随机抛物型方程的可观测性不等式。在直接的实际应用中，我们的重点集中在研究最佳执行器位置，以在随机抛物线方程的近似可控性背景下实现最小范数控制。我们引入了优化问题的综合表述，包括执行器位置的确定和相应的最小范数控制。更准确地说，我们将问题重新表述为两人零和博弈场景，从而推导出四个等价公式。最终，我们证实了关键结果，即松弛优化问题的解决方案可以作为经典问题的最佳执行器放置。