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, . We show that for a non-trivial , solutions to the Schrödinger equation, , satisfy . In particular, any set of positive Lebesgue measure can be used for observability. This leads to controllability with localization functions in and controls in . For continuous this follows from the results of Haraux [Har89] and Jaffard [Jaf90], while for (the rational torus) and this can be deduced from the results of Jakobson [Jak97].
[BG18] Stabilisation of wave equations on the torus with rough dampings (2018) (https://arxiv.org/abs/1801.00983)
[Bur] Wave control and second-microlocalization on geodesics (in preparation) | Zbl
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[Jaf90] Contrôle interne exact des vibrations d’une plaque rectangulaire, Port. Math., Volume 47 (1990) no. 4, pp. 423-429 | Zbl
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