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 , . On démontre que pour toute fonction non triviale, les solutions de l’équation de Schrödinger, , vérifient . 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 et des contrôles dans . Pour des continus, ce résultat est conséquence des travaux de Haraux [Har89] and Jaffard [Jaf90], tandis que pour (le tore rationnel) et on peut obtenir ce résultat comme conséquence des travaux de Jakobson [Jak97].
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].
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DOI : 10.5802/ahl.19
Mots clés : Schrödinger equation, control/observability for PDE, semiclassical analysis
@article{AHL_2019__2__331_0, author = {Burq, Nicolas and Zworski, Maciej}, title = {Rough controls for {Schr\"odinger} operators on 2-tori}, journal = {Annales Henri Lebesgue}, pages = {331--347}, publisher = {\'ENS Rennes}, volume = {2}, year = {2019}, doi = {10.5802/ahl.19}, zbl = {1421.35306}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/ahl.19/} }
Burq, Nicolas; Zworski, Maciej. Rough controls for Schrödinger operators on 2-tori. Annales Henri Lebesgue, Tome 2 (2019), pp. 331-347. doi : 10.5802/ahl.19. http://archive.numdam.org/articles/10.5802/ahl.19/
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