Rough controls for Schrödinger operators on 2-tori

Burq, Nicolas; Zworski, Maciej

doi:10.5802/ahl.19

Burq, Nicolas; Zworski, Maciej

Rough controls for Schrödinger operators on 2-tori

Annales Henri Lebesgue, Volume 2 (2019), pp. 331-347.

Metadata

DOI10.5802/ahl.19

Keywords Schrödinger 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, $𝕋^{2} = ℝ^{2} / ℤ \oplus γ ℤ$ . We show that for a non-trivial $W \in L^{\infty} (𝕋^{2})$ , solutions to the Schrödinger equation, $(i \partial_{t} + Δ) u = 0$ , satisfy ${∥ u |}_{t = 0} ∥_{L^{2} (𝕋^{2})} \leq K_{T} {∥ W u ∥}_{L^{2} ([0, T] \times 𝕋^{2})}$ . 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 $L^{4} ([0, T] \times 𝕋^{2})$ . For continuous $W$ this follows from the results of Haraux [Har89] and Jaffard [Jaf90], while for $𝕋^{2} = ℝ^{2} / {(2 π ℤ)}^{2}$ (the rational torus) and $T > π$ this can be deduced from the results of Jakobson [Jak97].

References

[AFKM15] Anantharaman, Nalini; Fermanian-Kammerer, Clotilde; Macià, Fabricio Semiclassical completely integrable systems: long-time dynamics and observability via two-microlocal Wigner measures, Am. J. Math., Volume 137 (2015) no. 3, pp. 577-638 | DOI | MR | Zbl

[AJM12] Aïssiou, Tayeb; Jakobson, Dmitry; Macià, Fabricio Uniform estimates for the solutions of the Schrödinger equation on the torus and regularity of semiclassical measures, Math. Res. Lett., Volume 19 (2012) no. 3, pp. 589-599 | DOI | Zbl

[AM14] Anantharaman, Nalini; Macià, Fabricio Semiclassical measures for the Schrödinger equation on the torus, J. Eur. Math. Soc., Volume 16 (2014) no. 6, pp. 1253-1288 | DOI | Zbl

[BBZ13] Bourgain, Jean; Burq, Nicolas; Zworski, Maciej Control for Schrödinger operators on 2-tori: rough potentials, J. Eur. Math. Soc., Volume 15 (2013) no. 5, pp. 1597-1628 | DOI | Zbl

[BD18] Bourgain, Jean; Dyatlov, Semyon Spectral gaps without the pressure condition, Ann. Math., Volume 187 (2018) no. 3, pp. 825-867 | DOI | MR | Zbl

[BG96] Burq, Nicolas; Gérard, Patrick Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes, C. R. Math. Acad. Sci. Paris, Volume 325 (1996) no. 7, pp. 749-752 | DOI | Zbl

[BG18] Burq, Nicolas; Gérard, Patrick Stabilisation of wave equations on the torus with rough dampings (2018) (https://arxiv.org/abs/1801.00983)

[BLR92] Bardos, Claude; Lebeau, Gilles; Rauch, Jeffrey Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optimization, Volume 30 (1992) no. 5, pp. 1024-1065 | DOI | MR | Zbl

[Bur] Burq, Nicolas Wave control and second-microlocalization on geodesics (in preparation) | Zbl

[BZ12] Burq, Nicolas; Zworski, Maciej Control for Schrödinger operators on tori, Math. Res. Lett., Volume 19 (2012) no. 2, pp. 309-324 | DOI | Zbl

[DJ18] Dyatlov, Semyon; Jin, Long Semiclassical measures on hyperbolic surfaces have full support, Acta Math., Volume 220 (2018) no. 2, pp. 297-339 | DOI | MR | Zbl

[Har89] Haraux, Alain Séries lacunaires et contrôle semi-interne des vibrations d’une plaque rectangulaire, J. Math. Pures Appl., Volume 68 (1989) no. 4, pp. 457-465 | Zbl

[Ing36] Ingham, Albert E. Some trigonometrical inequalities with applications to the theory of series, Math. Z., Volume 41 (1936), pp. 367-379 | DOI | MR | Zbl

[Jaf90] Jaffard, Stéphane Contrôle interne exact des vibrations d’une plaque rectangulaire, Port. Math., Volume 47 (1990) no. 4, pp. 423-429 | Zbl

[Jak97] Jakobson, Dmitry Quantum limits on flat tori, Ann. Math., Volume 145 (1997) no. 2, pp. 235-266 | DOI | MR | Zbl

[Jin17] Jin, Long Control for Schrödinger equation on hyperbolic surfaces (2017) (https://arxiv.org/abs/1707.04990, to appear in Math. Res. Lett.) | Zbl

[Kom92] Komornik, Vilmos On the exact internal controllability of a Petrowsky system, J. Math. Pures Appl., Volume 71 (1992) no. 4, pp. 331-342 | MR | Zbl

[Lio88] Lions, Jacques-Louis Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 2: Perturbations, Recherches en Mathématiques Appliquées, 9, Masson, 1988 | Zbl

[Mac09] Macià, Fabricio Semiclassical measures and the Schrödinger flow on Riemannian manifolds, Nonlinearity, Volume 22 (2009) no. 5, pp. 1003-1020 | DOI | MR | Zbl

[Zwo12] Zworski, Maciej Semiclassical analysis, Graduate Studies in Mathematics, 138, American Mathematical Society, 2012 | MR | Zbl

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