Burq, Nicolas; Zworski, Maciej
Rough controls for Schrödinger operators on 2-tori
Annales Henri Lebesgue, Volume 2 (2019), p. 331-347

Metadata

KeywordsSchrödinger equation, control/observability for PDE, semiclassical analysis

Abstract

The purpose of this note is to use the results and methods of [BBZ13] and [BZ12] to obtain control and observability by rough functions and sets on 2-tori, 𝕋 2 = 2 /γ. We show that for a non-trivial WL (𝕋 2 ), solutions to the Schrödinger equation, (i t +Δ)u=0, satisfy u| t=0 L 2 (𝕋 2 ) K T Wu L 2 ([0,T]×𝕋 2 ) . In particular, any set of positive Lebesgue measure can be used for observability. This leads to controllability with localization functions in L 2 (𝕋 2 ) and controls in L 4 ([0,T]×𝕋 2 ). For continuous W this follows from the results of Haraux [Har89] and Jaffard [Jaf90], while for 𝕋 2 = 2 /(2π) 2 (the rational torus) and T>π this can be deduced from the results of Jakobson [Jak97].


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