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

الأصل · EN

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 L²(M) can be stirred to 0 in an arbitrarily small time T by applying a suitable control g in L²([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≥ 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≤ alpha L² for some alpha < 2. Moreover, this bound implies C≤ alpha LOmega² where LOmega 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.