Nonlinear exact control below energy regularity

PDF p. 21: §5.2(B), below-energy and L² discussion

Assume the Axiom of Choice. Let $n\in \{2,3\}$ and let $\mathrm{\Omega }={\mathbb{R}}^n\setminus \mathrm{\Theta }$ be the connected exterior of a nonempty compact smooth obstacle, with geometry as in Exterior and truncated obstacle domains and non-trapping in the complete-ray sense of Non-trapping obstacle. Fix $T>0$ and $0\le s<1$. The Dirichlet spectral space $H_D^s(\mathrm{\Omega })$ is defined by Dirichlet spectral Sobolev scale and duality; in particular $H_D^0(\mathrm{\Omega })=L^2(\mathrm{\Omega })$.

For the cubic internally controlled system of Linear and nonlinear internally controlled exterior systems, with $g(v)=\lambda \mathrm{\mid }v{\mathrm{\mid }}^{2}v$ and $\lambda \in \mathbb{R}\setminus \{0\}$, determine for which such $s$ one can extend the article's nonlinear exact-control results to $H_D^s(\mathrm{\Omega })$. In particular, can one prove local exact controllability near zero: is there $\delta >0$ such that every pair $v_0,v_T\in H_D^s(\mathrm{\Omega })$ with norms at most $\delta$ can be joined in time $T$ by an admissible solution of the controlled equation? Here local exact controllability means endpoint reachability as defined in Reachability and observation vocabulary for controlled evolutions; the admissible control and solution classes must be specified, and the endpoint conditions must hold in $H_D^s(\mathrm{\Omega })$.

One proposed route is a Cauchy theory with forcing and nonlinear estimates in a suitable space $X_T\subset C([0,T];H_D^s(\mathrm{\Omega }))$, where $C([0,T];H_D^s(\mathrm{\Omega }))$ denotes norm-continuous paths as in Continuous and mixed time-space function spaces. The needed estimates and the existence of such a space are part of the problem, not assumptions already established here. Other suitable solution and control spaces are not excluded.

In the frozen v2 article, §5.2(B) lists below-energy nonlinear control as an open direction. It contrasts this with the article's linear result over $s\in [-2,2]$ and its nonlinear results at energy regularity and above, and identifies the two-dimensional cubic $L^2$ problem as mass-critical. The authors state that their reference [9] supplies no Cauchy theory at that level. These are descriptions of the article's claims and its stated state of knowledge, not independent verification of the cited proof or a claim that the question remains open throughout the later literature as of 2026-11-13. The linear result alone does not solve the nonlinear question.

The article is cited for attribution of the question and its stated context. The cited definitions supply the model, geometry, spectral scale, time-space notation and control terminology; none asserts the desired low-regularity control result. Choice is retained from the scale and system definitions The Axiom of Choice.