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

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].

On se propose dans cette note d’utiliser les résultats et les méthodes de [BBZ13] et [BZ12] pour obtenir des résultats de contrôle et de stabilisation par des fonctions de localisation peu régulières sur les tores de dimension 2, 𝕋 2 = 2 /γ. On démontre que pour toute fonction WL (𝕋 2 ) non triviale, les solutions de l’équation de Schrödinger, (i t +Δ)u=0, vérifient u| t=0 L 2 (𝕋 2 ) K T Wu L 2 ([0,T]×𝕋 2 ) . En particulier tout ensemble de mesure de Lebesgue strictement positive suffit pour l’observabilité. On en déduit des résultats de contrôlabilité exacte par des fonctions de localisation dans L 2 (𝕋 2 ) et des contrôles dans L 4 ([0,T]×𝕋 2 ). Pour des W continus, ce résultat est conséquence des travaux de Haraux [Har89] and Jaffard [Jaf90], tandis que pour 𝕋 2 = 2 /(2π) 2 (le tore rationnel) et T>π on peut obtenir ce résultat comme conséquence des travaux de Jakobson [Jak97].

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DOI: 10.5802/ahl.19
Classification: 35Q41,  35Q93,  49J20,  81Q93
Keywords: Schrödinger equation, control/observability for PDE, semiclassical analysis
Burq, Nicolas 1; Zworski, Maciej 2

1 Université Paris Sud, Université Paris-Saclay CNRS Mathématiques Bât. 307 91405 Orsay Cedex (France)
2 Mathematics Department University of California Berkeley, CA 94720 (USA)
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Burq, Nicolas; Zworski, Maciej. Rough controls for Schrödinger operators on 2-tori. Annales Henri Lebesgue, Volume 2 (2019), pp. 331-347. doi : 10.5802/ahl.19. http://archive.numdam.org/articles/10.5802/ahl.19/

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