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

Abstract

Given a control region $Ω$ 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 stirred to 0 in an arbitrarily small time $T$ by applying a suitable control $g$ in $L^2([0,T]xOmega)$, and, as $T$ tends to 0, the norm of $g$ grows like $e^(C/T)$ times the norm of the data. We investigate how $C$ depends on the geometry of $Omega$. We prove $C\geq d^{2}/4$ where $d$ is the largest distance of a point in $M$ from $Ω$. When $M$ is a segment of length $L$ controlled at one end, we prove $C\leq alpha L^{2}$ for some $alpha < 2$. Moreover, this bound implies $C\leq alpha L_{Omega}^2$ where $L_{Omega}$ is the length of the longest generalized geodesic in $M$ which does not intersect $Ω$. The control transmutation method used in proving this last result is of a broader interest.