An observability for parabolic equations from a measurable set in time

TL;DR

This paper establishes a new observability estimate for parabolic equations in convex domains using partial time-space observations, with applications to control problems, based on quantitative unique continuation at a single time point.

Contribution

It introduces a novel observability estimate for parabolic equations from partial time-space measurements, leveraging unique continuation at one time point.

Findings

New observability estimate for parabolic equations from partial measurements.

Application to bang-bang property in control problems.

Utilizes quantitative unique continuation at a single time.

Abstract

This paper presents a new observability estimate for parabolic equations in $\Omega\times(0,T)$, 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 $(0,T)$. 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.