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 Ω . 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]×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 ⩾ 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⩽α ∗ L 2 for some α ∗ <2 . Moreover, this bound implies C⩽α ∗ L Ω 2 where L Ω 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.