Question A: general boundary exact control

PDF p. 1: Question A

Assume countable choice. The article poses the following schematic question: let ${\mathrm{\Omega }}_0\subset {\mathbb{R}}^n$ be a bounded domain, $n\ge 1$, with boundary $\mathrm{\Gamma }$, let $\lambda \in \mathbb{R}\setminus \{0\}$, and let $s\ge 0$ and $T>0$. For each pair $u_0,u_1\in H^s({\mathrm{\Omega }}_0)$, can a full-boundary control $h$ produce a solution $u\in C([0,T];H^s({\mathrm{\Omega }}_0))$ of $i{u}_t+\mathrm{\Delta }u+\lambda \mathrm{\mid }u{\mathrm{\mid }}^{2}u=0$ on ${\mathrm{\Omega }}_0\times (0,T)$, with $u(0)=u_0$ and $u(T)=u_1$, using either Dirichlet data $u=h$ or Neumann data ${\mathrm{\partial }}_{\nu }u=h$ on $\mathrm{\Gamma }\times (0,T)$? This records the original broad question; its Neumann notation alone does not supply a normal on an arbitrary bounded domain.

For a precise instance with classical boundary geometry in dimension $n\ge 2$, additionally take ${\mathrm{\Omega }}_0$ to be a bounded $C^1$ domain. Its outward unit normal is the graph normal $(-Dh,1)\mathrm{/}\sqrt{1+\mathrm{\mid }Dh{\mathrm{\mid }}^2}$ in an oriented chart where the domain lies below the graph $z=h(y)$; the chart function here is unrelated to the boundary control. For a state continuously differentiable up to the boundary, ${\mathrm{\partial }}_{\nu }u={\mathrm{\nabla }}_{x}u\cdot \nu$, applied to its real and imaginary parts, and Dirichlet data are classical boundary restriction. At each $s$, fix in advance a complex Banach control space $U$ and an admissible solution/control class $\mathcal{A}\subseteq C([0,T];H^s({\mathrm{\Omega }}_0))\times U$. This class must specify a meaning for the nonlinear equation and the chosen boundary equality, including any weak traces, and must agree with these classical meanings whenever the state is sufficiently regular. For each such fixed formulation the question is whether every endpoint pair admits $(u,h)\in \mathcal{A}$ with the prescribed endpoints. The boundary regularity and fixed interpretation are explicit conditions on this precise instance, not extra hypotheses attributed to the original question. A formulation on a nonsmooth bounded domain must instead supply its own boundary structure and boundary interpretation before applying the endpoint quantifiers. No classical or weak normal trace for arbitrary Sobolev states is asserted here.

For $s\ge 0$ and any nonempty open $D\subseteq {\mathbb{R}}^n$, use the canonical $L^2$ realization ${\iota }_s:H^s({\mathbb{R}}^n)\to L^2({\mathbb{R}}^n)$ and let $R_{D}F=({\iota }_{s}F){\mathrm{\mid }}_D$, with restrictions identified up to almost-everywhere equality. Define the restriction space and its norm by $H^s(D)=R_D{H}^s({\mathbb{R}}^n),\mathrm{\parallel }u{\mathrm{\parallel }}_{H^s(D)}=\inf \{\mathrm{\parallel }F{\mathrm{\parallel }}_{H^s({\mathbb{R}}^n)}:R_{D}F=u\}\mathrm{.}$ Here the whole-space space is the Bessel-potential completion, identified with tempered distributions whose Fourier transform multiplied by $(1+\mathrm{\mid }\xi {\mathrm{\mid }}^2{)}^{s\mathrm{/}2}$ is a regular $L^2$ distribution, using the negative-sign $2\pi$ Fourier convention. The restriction-space norm is definite and complete by the justification below. The path space means continuity in this norm. This restriction-space convention does not impose homogeneous boundary conditions. The article's later smooth truncated obstacle domain is one instance of the boundary geometry described above.

Cubic NLS and boundary control formulations

Continuous and mixed time-space function spaces

Reachability and observation vocabulary for controlled evolutions

Real-order Bessel-potential completion Hˢ

Real-order Hˢ as weighted Fourier distributions

The Axiom of Countable Choice (AC_ω)

Bounded C1 domains and their outward normals

Classical normal derivative

Restriction Bessel-potential spaces on polyhedral domains

A quotient of a Banach space by a closed subspace is Banach

Complex Holder, Minkowski, and the quotient norm

Integral over a measurable subset

Monotonicity and nonnegative homogeneity of the nonnegative integral

Only the formulation and its spaces are justified here. For every $s\ge 0$, Restriction Bessel-potential spaces on polyhedral domains supplies the linear injection ${\iota }_s$ with $\mathrm{\parallel }{\iota }_{s}F{\mathrm{\parallel }}_{L^2({\mathbb{R}}^n)}\le \mathrm{\parallel }F{\mathrm{\parallel }}_{H^s({\mathbb{R}}^n)}$ and the restriction-space convention used above; despite its title, that definition applies to every nonempty open $D$, not only polyhedra. Restriction of an $L^2$ class is well-defined, since a null disagreement set remains null on $D$. The complex $L^2$ norm and vector operations are supplied by Complex Holder, Minkowski, and the quotient norm. By the subset integral convention of Integral over a measurable subset and monotonicity in Monotonicity and nonnegative homogeneity of the nonnegative integral, $\mathrm{\parallel }{R}_{D}F{\mathrm{\parallel }}_{L^2(D)}^2={\int }_D\mathrm{\mid }{\iota }_{s}F{\mathrm{\mid }}^2\le {\int }_{{\mathbb{R}}^n}\mathrm{\mid }{\iota }_{s}F{\mathrm{\mid }}^2\le \mathrm{\parallel }F{\mathrm{\parallel }}_{H^s({\mathbb{R}}^n)}^2\mathrm{.}$ Thus $R_D$ is linear and bounded. Its kernel $M_D$ is a closed linear subspace: if $R_{D}F\mathrm{\ne }0$, the ball of radius $\mathrm{\parallel }{R}_{D}F{\mathrm{\parallel }}_{L^2(D)}\mathrm{/}2$ about $F$ misses $M_D$, since $G\in M_D$ would imply $\mathrm{\parallel }{R}_{D}F{\mathrm{\parallel }}_{L^2(D)}=\mathrm{\parallel }{R}_D(F-G){\mathrm{\parallel }}_{L^2(D)}\le \mathrm{\parallel }F-G{\mathrm{\parallel }}_{H^s}$.

The whole-space space is Banach by Real-order Bessel-potential completion Hˢ. Two extensions have the same restriction exactly when their difference lies in $M_D$, so the range is linearly identified with $H^s({\mathbb{R}}^n)\mathrm{/}{M}_D$ and the displayed infimum is precisely its quotient norm. Under countable choice, A quotient of a Banach space by a closed subspace is Banach makes this a definite Banach norm for every $s\ge 0$, without boundary regularity. Countable choice is the assumption of The Axiom of Countable Choice (AC_ω) used by the completion and quotient suppliers. The Fourier description and normalization are those of Real-order Hˢ as weighted Fourier distributions. This establishes the Banach-target hypothesis of Continuous and mixed time-space function spaces.

For the conditional $C^1$ instance with $n\ge 2$, Bounded C1 domains and their outward normals supplies the normal and Classical normal derivative supplies its classical derivative convention. These definitions make no weak-trace assertion. The article-specific instance and the complex cubic notation are those of Cubic NLS and boundary control formulations; that supplier is not applied to an arbitrary nonsmooth domain. For a fixed admissible class, the endpoint relation is $(u_0,h,u_1)\in {\mathcal{R}}_T$ exactly when some $(u,h)\in \mathcal{A}$ has these endpoints. The predicate is therefore exact controllability on $H^s({\mathrm{\Omega }}_0)$ in Reachability and observation vocabulary for controlled evolutions. Neither existence of such an admissible formulation at every regularity nor its controllability is inferred from this definition.

The frozen article poses this general question and reports local exact controllability on the whole boundary in regularity ranges depending on dimension, around smooth solutions. Rosier and Zhang's publisher abstract independently reports such local results for specified ranges, for both boundary conditions. Local exact controllability concerns endpoints sufficiently close to reference endpoints; it does not by itself answer the question for arbitrary endpoint pairs. As of 2026-11-13, the sources consulted here do not establish a general answer for all stated parameters; this limited search is not a comprehensive current-literature certification.

The article citation attributes Question A and its context. The independent Rosier–Zhang abstract records only the partial result described there; no theorem from that paper is used as a logical premise. The boundary qualification and restriction-quotient justification are local formulation repairs, not claimed additions by the article's authors and not a solution to Question A.