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

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, ${𝕋}^{2}={ℝ}^{2}/ℤ\oplus \gamma ℤ$. We show that for a non-trivial $W\in {L}^{\infty }\left({𝕋}^{2}\right)$, solutions to the Schrödinger equation, $\left(i{\partial }_{t}+\Delta \right)u=0$, satisfy ${\parallel u|}_{t=0}{\parallel }_{{L}^{2}\left({𝕋}^{2}\right)}\le {K}_{T}{\parallel Wu\parallel }_{{L}^{2}\left(\left[0,T\right]×{𝕋}^{2}\right)}$. In particular, any set of positive Lebesgue measure can be used for observability. This leads to controllability with localization functions in ${L}^{2}\left({𝕋}^{2}\right)$ and controls in ${L}^{4}\left(\left[0,T\right]×{𝕋}^{2}\right)$. For continuous $W$ this follows from the results of Haraux [Har89] and Jaffard [Jaf90], while for ${𝕋}^{2}={ℝ}^{2}/{\left(2\pi ℤ\right)}^{2}$ (the rational torus) and $T>\pi$ this can be deduced from the results of Jakobson [Jak97].

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