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Our goal is to study controllability and observability properties of the 1Dheat equation with internal control (or observation) set$\omega_{\varepsilon}=(x_{0}-\varepsilon, x_{0}+\varepsilon )$, in the limit$\varepsilon\rightarrow 0$, where $x_{0}\in (0,1)$. It is known that dependingon arithmetic properties of $x_{0}$, there may exist a minimal time $T_{0}$ ofpointwise control at $x_{0}$ of the heat equation. Besides, for any$\varepsilon$ fixed, the heat equation is controllable with control set$\omega_{\varepsilon}$ in any time $T>0$. We relate these two phenomena. Weshow that the observability constant on $\omega_\varepsilon$ does not convergeto $0$ as $\varepsilon\rightarrow 0$ at the same speed when $T>T_{0}$ (in whichcase it is comparable to $\varepsilon^{1/2}$) or $T<T_{0}$ (in which case itconverges faster to $0$). We also describe the behavior of optimal $L^{2}$null-controls on $\omega_{\varepsilon}$ in the limit $\varepsilon \rightarrow0$.

From internal to pointwise control for the 1D heat equation and minimal control time