Observability Inequality from Measurable Sets and the Shape Design Problem for Stochastic Parabolic Equations

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.