Unique Continuation for Stochastic Heat Equations

We establish a unique continuation property for stochastic heat equations evolving in a bounded domain $G$. Our result shows that the value of the solution can be determined uniquely by means of its value on an arbitrary open subdomain of $G$ at any given positive time constant. Further, when $G$ is convex and bounded, we also give a quantitative version of the unique continuation property. As applications, we get an observability estimate for stochastic heat equations, an approximate result and a null controllability result for a backward stochastic heat equation

Observability Inequality of Backward Stochastic Heat Equations for Measurable Sets and Its Applications

This paper aims to provide directly the observability inequality of backward stochastic heat equations for measurable sets. As an immediate application, the null controllability of the forward heat equations is obtained. Moreover, an interesting relaxed optimal actuator location problem is formulated, and the existence of its solution is proved. Finally, the solution is characterized by a Nash equilibrium of the associated game problem

Observability inequalities for the backward stochastic evolution equations and their applications

The present article delves into the investigation of observability inequalities pertaining to backward stochastic evolution equations. We employ a combination of spectral inequalities, interpolation inequalities, and the telegraph series method as our primary tools to directly establish observability inequalities. Furthermore, we explore three specific equations as application examples: a stochastic degenerate equation, a stochastic fourth order parabolic equation and a stochastic heat equation. It is noteworthy that these equations can be rendered null controllability with only one control in the drift term to each system

Optimal Actuator Location of the Minimum Norm Controls for Stochastic Heat Equations

In this paper, we study the approximate controllability for the stochastic heat equation over measurable sets, and the optimal actuator location of the minimum norm controls. We formulate a relaxed optimization problem for both actuator location and its corresponding minimum norm control into a two-person zero sum game problem and develop a sufficient and necessary condition for the optimal solution via Nash equilibrium. At last, we prove that the relaxed optimal solution is an optimal actuator location for the classical problem