Global defocusing controllability in large time

PDF p. 22: §5.2(F)

Assume the Axiom of Choice, as in the controlled-system definition. In dimension two, for the defocusing cubic nonlinear Schrödinger equation in the exterior of a smooth compact obstacle, can one establish global controllability in sufficiently large time by combining stabilization with local controllability? The exterior is connected, and non-trapping has the complete-ray meaning of Non-trapping obstacle. In the internal formulation the equation is $i{\mathrm{\partial }}_{t}v+\mathrm{\Delta }v+\lambda \mathrm{\mid }v{\mathrm{\mid }}^{2}v=\phi h$, with zero Dirichlet condition on the obstacle, $\lambda <0$ in this sign convention, and control applied through the cutoff $\phi$ outside a ball containing the obstacle.

Here exact controllability on a specified set of admissible states means that each ordered pair of states in that set can be joined by an admissible controlled solution in the specified time. A complete result must specify the state and control spaces, solution and trace notions, the admissible data, the geometry and any additional hypotheses, and whether the required time is uniform over all data or depends on a bound for them. A possible extension of this question is global controllability for the outer-boundary Dirichlet formulation on ${\mathrm{\Omega }}_0=B_{R_0}(0)\setminus \mathrm{\Theta }$, with $\mathrm{\Theta }\subset B_{R_0}(0)$, $u=h$ on ${\mathrm{\Gamma }}_1=\mathrm{\partial }{B}_{R_0}(0)$ and $u=0$ on ${\mathrm{\Gamma }}_2=\mathrm{\partial }\mathrm{\Theta }$. This boundary-control extension is proposed here. The passage from an exterior internal control to this boundary control requires its own compatible extension and trace properties.

Linear and nonlinear internally controlled exterior systems

Exterior and truncated obstacle domains

Reachability and observation vocabulary for controlled evolutions

The Axiom of Choice

Cubic NLS and boundary control formulations

As of the frozen v2 article, Section 5.2(F) presents this as a proposed direction, not a proved global theorem. It proposes combining the exponential stabilization result cited there as [14] with the paper's local controllability results, in the spirit of [15,23], and explicitly says that the functional settings and geometric assumptions must be matched. Exponential stabilization means decay to zero at an exponential rate in a specified norm for a specified damped evolution; neither the damping equation nor the admissibility of its feedback as a control for the present system is established by that terminology.

The article's local internal result concerns sufficiently small endpoint data in $H_D^s(\mathrm{\Omega })$ for $s\in \{1,2\}$ and any prescribed positive time, under its stated assumptions. This is a local result and does not settle the global question. A limited title-and-topic search identified the cited exterior-domain stabilization paper, but its full theorem and hypotheses were not independently checked here; no exhaustive current literature or worldwide novelty review is claimed. In particular, this item records the authors' question and does not assert that it remains unsolved in all subsequent literature.

The article is the source of the question and of the descriptions of its own local result and proposed route. Its references [14], [15] and [23] are contextual comparisons, not imported premises or independently verified statements in this item. Matching damping, control support, solution spaces, time reversal or an appropriate return construction, and uniformity of decay on the relevant data sets remain parts of the proposed program.

The outer-boundary Dirichlet formulation is supplied by Cubic NLS and boundary control formulations; its global-controllability extension above is a proposal in this item and is not expressly stated in Section 5.2(F).