An observability estimate for parabolic equations from a measurable set in time and its applications

This paper presents a new observability estimate for parabolic equations in $\Omega \times \left(0,T\right)$ , where $\Omega$ is a convex domain. The observation region is restricted over a product set of an open nonempty subset of $\Omega$ and a subset of positive measure in $\left(0,T\right)$ . 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.