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How Violent are Fast Controls for Schrödinger and Plate Vibrations?

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Abstract.

Given a time T>0 and a region Ω on a compact Riemannian manifold M, we consider the best constant, denoted C T,Ω , in the observation inequality for the Schrödinger evolution group of the Laplacian Δ with Dirichlet boundary condition: We investigate the influence of the geometry of Ω on the growth of C T,Ω as T tends to 0.

By duality, C T,Ω is also the controllability cost of the free Schrödinger equation on M with Dirichlet boundary condition in time T by interior controls on Ω. It relates to hinged vibrating plates as well. We analyze separately the effects of wavelengths which are greater and lower than the order of the control time T. We emphasize a tool of wider scope: the control transmutation method.

We prove that C T,Ω grows at least like exp(d 2/4T), where d is the largest distance of a point in M from Ω, and at most like exp(α* L Ω 2/T), where L Ω is the length of the longest generalized geodesic in M which does not intersect Ω, and α* ∈]0,4[ is the best constant in the following inequality for the Schrödinger equation on the segment [0,L] observed from the left end: where A is the operator ∂ x 2 with domain D(A)={f∈H 2(0,L),|,Bf(0)=0=f(L)} and the inequality holds with B=1 and with B=∂ x . We also deduce such upper bounds on product manifolds for some control regions which are not intersected by all geodesics.

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  1. Centre de Mathématiques, École Polytechnique, UMR CNRS 7640, 91128, Palaiseau, France

    Luc Miller

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Communicated by P.-L. Lions

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Miller, L. How Violent are Fast Controls for Schrödinger and Plate Vibrations?. Arch. Rational Mech. Anal. 172, 429–456 (2004). https://doi.org/10.1007/s00205-004-0312-y

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