Prior energy-critical null control on Euclidean space

PDF p. 3: end of opening paragraph; PDF p. 5: paragraph after Corollary 1.6

The article attributes to Braz e Silva, Capistrano–Filho, Carvalho and Dos Santos Ferreira local null controllability in $H^1({\mathbb{R}}^3)$ for the energy-critical nonlinear Schrödinger equation, with controls supported outside a ball. It identifies the three-dimensional quintic equation as energy-critical and compares its own quintic result with that reported whole-space result.

For the mathematical conventions in this remark, assume the Axiom of Choice. The whole-space quintic equation is written in the template form $i{\mathrm{\partial }}_{t}u+\mathrm{\Delta }u+\lambda \mathrm{\mid }u{\mathrm{\mid }}^{4}u=F$ on ${\mathbb{R}}^3\times (0,T)$, where $u$ is complex-valued, $\mathrm{\Delta }$ is the spatial Laplacian, $\lambda \in \mathbb{R}\setminus \{0\}$ and $F$ denotes the control forcing. This uses the same quintic power as the internally controlled system, with no obstacle or boundary condition, and does not specify the admissible controls of reference [7]. Energy-critical refers to the unforced equation's scaling $u_{\rho }(t,x)={\rho }^{1\mathrm{/}2}u({\rho }^{2}t,\rho x)$, $\rho >0$, preserving the homogeneous first-order seminorm $\mathrm{\parallel }\mathrm{\nabla }u(t){\mathrm{\parallel }}_{L^2({\mathbb{R}}^3)}$ at corresponding times for sufficiently smooth states of finite seminorm. This is the gradient-norm scaling terminology of the critical-exponents remark; it supplies no controllability assertion.

Here $H^1({\mathbb{R}}^3)=W^{1,2}({\mathbb{R}}^3;\mathbb{C})$ consists of complex $L^2$ classes whose first weak spatial derivatives also belong to $L^2$. Null controllability means steering the initial state to zero at the prescribed terminal time; the local qualification restricts the initial states to a neighborhood of zero in the relevant state topology. The precise admissible controls, solution class, time and smallness hypotheses belong to the attributed result.

Reachability and observation vocabulary for controlled evolutions

Integer-order Sobolev spaces and their norms

Critical nonlinearities named in the article

Linear and nonlinear internally controlled exterior systems

The Axiom of Choice

The article lists reference [7] as P. Braz e Silva, R. de A. Capistrano–Filho, J. D. do N. Carvalho and D. Dos Santos Ferreira, Control of the Schrödinger equation in R3: the critical case, preprint, 2026. This remark records the article's attribution; the preprint has not been independently read here, and its reported result is not used as a premise.