Large-data controllability for the energy-critical quintic equation

Establish large-data controllability for the energy-critical quintic nonlinear Schrödinger equation in the exterior domain, including in the setting of \mathbb{R}^3, beyond the local controllability result near the zero solution.

Background

The paper proves local exact controllability for the quintic nonlinear Schrödinger equation near the zero solution in dimensions two and three. In dimension three, the quintic equation is energy-critical, making the extension from small data to arbitrary or large data substantially more difficult.

The authors suggest that a possible route could combine stabilization with local controllability, but no such global or large-data result is established. The unresolved issue remains significant even in the simpler whole-space setting, so it is also open for the exterior-domain and boundary-control configurations considered here.

References

Large-data controllability results in the energy-critical case remain open even on $\mathbb{R}3$ ; one possible approach would combine a stabilization mechanism with local controllability results, as in item (F) below.

— Nonlinear Schrödinger equation in an exterior domain  (2610.11968 - Filho et al., 8 Oct 2026) in Section 5, subsection “Open problems,” item (C)