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

Given a control region Omega on a compact Riemannian manifold M, we consider the heat equation with a source term g localized in Q. 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] x 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. We prove Cgreater than or equal tod(2)/4 where d is the largest distance of a point in M from Q. When M is a segment of length L controlled at one end, we prove Cless than or equal toalpha(*)L(2) for some alpha(*)<2. Moreover, this bound implies C less than or equal to alpha*L-Omega(2) where L-Omega is the length of the longest generalized geodesic in M which does not intersect Omega. The control transmutation method used in proving this last result is of a broader interest. (C) 2004 Elsevier Inc. All rights reserved.