How violent are fast controls for Schroedinger and plate vibrations ?
Abstract
Given a time T>0 and a region Omega on a compact Riemannian manifold M, we consider the best constant, denoted CT,Omega, in the observation inequality for the Schroedinger evolution group of the Laplacian Delta with Dirichlet boundary condition: for all f in L2(M), ||f||L2(M) ≤ CT,Omega ||exp(itDelta)f||L2((0,T)xOmega). We investigate the influence of the geometry of Omega on the growth of CT,Omega as T tends to 0. By duality, CT,Omega is also the controllability cost of the free Schroedinger equation on M with Dirichlet boundary condition in time T by interior controls on Omega. It relates to hinged vibrating plates as well. We emphasize a tool of wider scope: the control transmutation method. We prove that CT,Omega grows at least like exp(d2/4T), where d is the largest distance of a point in M from Omega, and at most like exp(alpha L2/T), where L is the length of the longest generalized geodesic in M which does not intersect Omega, and alpha is a constant in ]0,4[ (it is the growth rate of the controllability cost in a similar one dimensional problem). We also deduce such upper bounds on product manifolds for some control regions which are not intersected by all geodesics.
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