Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time

Earlier version on arXiv:math.AP/0307158Given a control region $\Omega$ on a compact Riemannian manifold $M$, we consider the heat equation with a source term $g$ localized in $\Omega$. It is known that any initial data in $L^{2}(M)$ can be steered to $0$ in an arbitrarily small time $T$ by applying a suitable control $g$ in $L^{2}([0,T]\times\Omega)$, and, as $T$ tends to $0$, the norm of $g$ grows like $\exp(C/T)$ times the norm of the data. We investigate how $C$ depends on the geometry of $\Omega$. %% 72 words We prove $C\geq d^{2}/4$ where $d$ is the largest distance of a point in $M$ from $\Omega$. When $M$ is a segment of length $L$ controlled at one end, we prove $C\leq \alpha_{*}L^{2}$ for some $\alpha_{*}