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

Given a control region Ω on a compact Riemannian manifold M , we consider the heat equation with a source term g localized in Ω. It is known that any initial data in L2(M) can be stirred to 0 in an arbitrarily small time T by applying a suitable control g in L2([0, T ]×Ω), 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 Ω. We prove C ≥ d2/4 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 ≤ α∗L for some α∗ < 2. Moreover, this bound implies C ≤ α∗LΩ where LΩ 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.