Asymptotic observability identity for the heat equation in R^d

TL;DR

This paper establishes an asymptotic observability identity for the heat equation in orm space, enabling approximate solution recovery from lattice point observations and extending classical sampling theorems.

Contribution

It introduces an asymptotic observability identity for the heat equation in orm space, extending sampling theory and providing controllability results.

Findings

Approximate solution recovery from lattice observations.

Extension of Shannon-Whittaker sampling theorem.

Weak observability identity with finitely many points.

Abstract

We build up an asymptotic observability identity for the heat equation in the whole space. It says that one can approximately recover a solution, through observing it over some countable lattice points in the space and at one time. This asymptotic identity is a natural extension of the well-known Shannon-Whittaker sampling theorem \cite{Shannon,Whittaker}. According to it, we obtain a kind of feedback null approximate controllability for impulsively controlled heat equations. We also obtain a weak asymptotic observability identity with finitely many observation lattice points. This identity holds only for some solutions to the heat equation.