A deterministic optimal design problem for the heat equation

For the heat equation on a bounded subdomain $\mathrm{\Omega }$ 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 $\mathrm{\Omega }$ in previous works. The explicit nature of the heat packets gives new information about the observability constant in the inverse problem.