Large-data control for energy-critical NLS

PDF p. 21: §5.2(C), large-data discussion; PDF p. 5: Corollary 1.6

Assume the Axiom of Choice and the geometry and cutoff of Linear and nonlinear internally controlled exterior systems, in dimension three, with a non-trapping obstacle. For the internally controlled Dirichlet equation $i{\mathrm{\partial }}_{t}v+\mathrm{\Delta }v+\lambda \mathrm{\mid }v{\mathrm{\mid }}^{4}v=\phi h$, can one obtain controllability for large initial data in the energy space $H_D^1(\mathrm{\Omega })=H_0^1(\mathrm{\Omega })$? In particular, determine conditions under which a control $h$ drives a prescribed initial state to zero, or connects prescribed initial and terminal states, in a suitable solution class and time. Here large data means that no smallness restriction on the energy-space norm is imposed. Exact controllability between two states means that the solution takes those states at the initial and terminal times; null controllability is the special case with zero terminal state.

The equation, boundary condition $v{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}=0$, control multiplier $\phi$, and domain are those of Linear and nonlinear internally controlled exterior systems. The energy-space notation is that of Dirichlet spectral Sobolev scale and duality; the Axiom of Choice is retained as required by those formulations. The question asks which control and solution spaces, times, signs of $\lambda$, and additional hypotheses permit a result; it does not assert that every such choice works.

The frozen article, §5.2(C), reports large-data control in the energy-critical case as open even on ${\mathbb{R}}^3$ and proposes combining stabilization with local controllability. Its Corollary 1.6 states a small-data result near zero in the exterior setting. These are attributed statements, not proofs supplied by this item.

In the independently consulted whole-space paper, Theorem 1.1 states null controllability for sufficiently small $H^1({\mathbb{R}}^3)$ initial data with a cutoff control, and §4(B) identifies large-data control as an open issue for that system. This is a partial result and source-specific status, not a proof for exterior domains or a certification that no later solution exists. The source review was limited to the theorem statement and the stated open issue, consulted on 2026-11-13. No proof or exhaustive current-literature review is claimed.

The frozen article is the source of the question and its suggested stabilization approach. The whole-space reference provides context for the reported partial result and open status; its theorem is not used as a premise for an exterior-domain deduction.