An observability for parabolic equations from a measurable set in time

Abstract

This paper presents a new observability estimate for parabolic equations in $Ω\times(0,T)$, where $Ω$ is a convex domain. The observation region is restricted over a product set of an open nonempty subset of $Ω$ and a subset of positive measure in $(0,T)$. This estimate is derived with the aid of a quantitative unique continuation at one point in time. Applications to the bang-bang property for norm and time optimal control problems are provided.