Our goal is to study controllability and observability properties of the 1D heat equation with internal control (or observation) set $omega_{varepsilon}=(x_{0}-varepsilon, x_{0}+varepsilon )$, in the limit $varepsilonrightarrow 0$, where $x_{0}in (0,1)$. It is known that depending on arithmetic properties of $x_{0}$, there may exist a minimal time $T_{0}$ of pointwise 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. We show that the observability constant on $omega_varepsilon$ does not converge to $0$ as $varepsilonrightarrow 0$ at the same speed when $T>T_{0}$ (in which case it is comparable to $varepsilon^{1/2}$) or $T<T_{0}$ (in which case it converges faster to $0$). We also describe the behavior of optimal $L^{2}$ null-controls on $omega_{varepsilon}$ in the limit $varepsilon rightarrow 0$.
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