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

TL;DR

This paper establishes observability inequalities for backward stochastic heat equations on measurable sets, enabling null controllability results, formulating an optimal actuator placement problem, and linking solutions to Nash equilibria in game theory.

Contribution

It provides the first direct observability inequality for backward stochastic heat equations on measurable sets and applies it to control and optimization problems.

Findings

Null controllability of forward heat equations achieved.

Existence of optimal actuator location proved.

Solution characterized as Nash equilibrium in a game setting.

Abstract

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.