Nonlinear Schrödinger equation in an exterior domain
We study the exact controllability of the cubic nonlinear Schrödinger equation in the exterior $Ω=\mathbb{R}^n\setminusΘ$ of a non-trapping obstacle $Θ$, with internal controls supported outside a large ball, and the corresponding boundary control problem on $Ω_0=B_{R_0}\setminusΘ$ with Dirichlet controls acting only on the outer sphere $\partial B_{R_0}$. Using the local smoothing effect of Burq, Gérard and Tzvetkov [9], we prove observability inequalities for the linear Schrödinger equation in the whole scale of Sobolev spaces $H^σ_D(Ω)$, $σ\in[-2,2]$, associated with the Dirichlet Laplacian; they require only the non-trapping assumption (no star-shapedness of $Θ$), and their proofs use no normal boundary traces. Combined with Strichartz-type estimates and a perturbation argument, this yields local exact controllability in $H^1_0(Ω)$ and in $H^2(Ω)\cap H^1_0(Ω)$, in dimensions two and three, around the zero solution and, for internal controls and under a unique continuation assumption, around nontrivial trajectories. The same method applies to the quintic equation, which is energy-critical in dimension three. Several open problems are discussed.