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 C_{T,Omega}, in the observation inequality for the Schroedinger evolution group of the Laplacian Delta with Dirichlet boundary condition: for all f in L^2(M), ||f||_{L^2(M)} \leq C_{T,Omega} ||exp(itDelta)f||_{L^2((0,T)xOmega)}. We investigate the influence of the geometry of Omega on the growth of C_{T,Omega} as T tends to 0. By duality, C_{T,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 C_{T,Omega} grows at least like exp(d^2/4T), where d is the largest distance of a point in M from Omega, and at most like exp(alpha L^2/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.

23 pages, a4paper, no figures

References in corpus (1)

• Control in the presence of a black box

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