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Article Dans Une Revue SIAM Journal on Mathematical Analysis Année : 2010

Global Controllability and Stabilization for the Nonlinear Schrödinger Equation on Some Compact Manifolds of Dimension 3

Camille Laurent

Résumé

We prove global internal controllability in large time for the nonlinear Schrödinger equation on some compact manifolds of dimension $3$. The result is proved under some geometrical assumptions : geometric control and unique continuation. We give some examples where they are fulfilled on $\Tot$, $S^3$ and $S^2\times S^1$. We prove this by two different methods both inherently interesting. The first one combines stabilization and local controllability near $0$. The second one uses successive controls near some trajectories. We also get a regularity result about the control if the data are assumed smoother. If the $H^1$ norm is bounded, it gives a local control in $H^1$ with a smallness assumption only in $L^2$. We use Bourgain spaces.
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Dates et versions

hal-00366912 , version 1 (10-03-2009)

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Camille Laurent. Global Controllability and Stabilization for the Nonlinear Schrödinger Equation on Some Compact Manifolds of Dimension 3. SIAM Journal on Mathematical Analysis, 2010, 42 (2), pp.785 - 832. ⟨10.1137/090749086⟩. ⟨hal-00366912⟩
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