A deterministic optimal design problem for the heat equation

TL;DR

This paper introduces a deterministic approach to optimizing the shape and placement of observation domains for the heat equation, improving understanding of observability constants without relying on randomness or geometric assumptions.

Contribution

It presents a novel decomposition of heat functions into heat packets, eliminating the need for randomization and geometric constraints in observability analysis.

Findings

Explicit heat packet decomposition enhances observability constant analysis

Removal of randomization simplifies the optimal design problem

Provides new insights into inverse problem observability constants

Abstract

For the heat equation on a bounded subdomain $\Omega$ of $\mathbb{R}^d$, we investigate the optimal shape and location of the observation domain in observability inequalites. A new decomposition of $L^2(\mathbb{R}^d)$ into heat packets allows us to remove the randomisation procedure and assumptions on the geometry of $\Omega$ in previous works. The explicit nature of the heat packets gives new information about the observability constant in the inverse problem.