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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Prior full and partial boundary control results reported in the introduction

PDF p. 1: final paragraph; PDF p. 2: §1.2, linear and higher-dimensional boundary results

Statement

Assume countable choice for the function-space conventions used here. The article reports three earlier boundary-control results. Rosier and Zhang obtained local exact control for the cubic equation with Dirichlet or Neumann control acting on the whole boundary, around a smooth reference solution, for dimension-dependent Sobolev exponents. Local exact control means that initial and final states sufficiently near the reference endpoints can be joined by an admissible controlled solution.

For the linear equation, the article attributes partial-boundary control under a multiplier condition to Machtyngier. In that setting the controlled part is selected by the sign of (x−x0)⋅ν(x), with x0 fixed and ν the outward unit normal. The cited theorem uses space-time L2 Dirichlet controls and initial data in H−1. For an open domain D, our notation H−1(D) denotes bounded conjugate-linear functionals on H01(D;C), with the operator norm and evaluation pairing linear in the functional and conjugate-linear in the test vector.

The article also reports that, for certain disjoint smooth strictly convex obstacles satisfying additional geometric conditions, Burq's plate-equation result and Burq–Zworski's Schrödinger result yield outer-boundary control despite trapped rays. The Schrödinger theorem concerns D=De∖Θ, with De smooth and bounded and containing the convex hull of the obstacles; its additional obstacle conditions remain those of the cited §6.2. In the latter source, for each T>0 and ε>0, initial data in its H01+ε class can be driven to zero with a space-time L2 Dirichlet control on the outer boundary. Here H01+ε reproduces the notation printed in that cited theorem; this contextual report does not identify it with either a closure of compactly supported smooth functions or a Dirichlet spectral-power domain.

External input

The following are imported theorem statements under the separate hypotheses recorded in their citation contracts; no local proof is claimed.

Rosier–Zhang, Theorem 1.1, supplies the full-boundary nonlinear control attribution.

Machtyngier, Theorem 1.1, supplies the linear multiplier-boundary control attribution.

Burq–Zworski, Theorem 1, supplies the outer-boundary Schrödinger control attribution under its §6.2 obstacle conditions.

Burq, Theorem 0.1, supplies only the plate-equation comparison below, under its own H1 and H2 hypotheses.

Source notes

These are background attributions and do not supply a proof of the present article's control results. The three earlier results have different equations, geometries, data classes and control requirements; no implication between them is asserted. The original article is cited for its historical account, and the independent sources record the corresponding earlier theorem statements. The notation H01+ε in the preceding historical statement is kept literally as printed in Burq–Zworski, Theorem 1, for every ε>0. The inspected passages do not explicitly define its all-order convention. We therefore make no equality claim between its higher-order boundary compatibility requirements and any spectral or closure convention used elsewhere in this collection. No result here uses that data class as an operational premise.

For comparison, Burq's original plate theorem, Theorem 0.1, states its displacement class explicitly as H1+ε(Ω)∩H01(Ω), with velocity class H−1+ε(Ω), for every ε>0. This is a separate plate-equation theorem under its own hypotheses; it is not used to define Burq–Zworski's notation or to transfer a control theorem between equations. Its statement and geometric hypotheses were read in the publisher sample and archival text extractions, whose formula OCR is imperfect, without reconstructing the proof.

Machtyngier uses real inner products on complex function spaces. The present conjugate-dual convention specifies our notation; the historical theorem remains reported with its source's real-pairing convention. No solution map is transported between pairing conventions and no equality of the source's norm normalization with our operator norm is asserted.

Depends on

Dependency tree · 2 levels

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External dependencies

Exact boundary controllability of the nonlinear Schrödinger equation · Theorem 1.1, journal page 4131

For every $T>0$, endpoints sufficiently near $w(0)$ and $w(T)$ in $H^s(D)$ can be joined by a solution in $C([0,T];H^s(D))$ of the cubic NLS, using full-boundary Dirichlet or Neumann control.

Hypotheses: $D\subset\mathbb R^n$ is a bounded domain. The nonzero constant $\lambda$ may be complex. A smooth reference solution $w$ exists on $D_1\times(-\eta,T+\eta)$ for a bounded neighborhood $D_1$ containing the closure of $D$, with $\eta>0$. The exponent satisfies $s>n/2$, or $0\le s<n/2$ with $1\le n<2+2s$, or $n=2$ with $s\in\{0,1\}$. Both boundary prescriptions act on the whole boundary; the theorem promises an appropriate control, not a specified common boundary norm.

Reading limits: Read the accessible full-paper text extraction of equations (1.1)–(1.3) and Theorem 1.1, printed pp.4129–4131. Fraction bars and some relation glyphs are imperfect in the extraction; the exponent alternatives were checked against the abstract in the same source. No proof reconstruction or additional boundary-control regularity is claimed.

Exact Controllability for the Schrödinger Equation · Machtyngier (1994), printed pp.24–25, equations (1.1), (1.3), Theorem 1.1 and real-inner-product conventions at the end of §1

For every $T>0$ and $y^0\in H^{-1}(D)$, there is $v\in L^2(\Gamma_+\times(0,T))$ such that the transposition solution of $iy_t+\Delta y=0$, with $y=v$ on $\Gamma_+$ and $y=0$ elsewhere on the boundary, lies in $C([0,T];H^{-1}(D))$ and satisfies $y(T)=0$.

Hypotheses: $D\subset\mathbb R^n$ is open and bounded with $C^3$ boundary. Fix $x_0\in\mathbb R^n$ and put $\Gamma_+=\{x\in\partial D:(x-x_0)\cdot\nu(x)>0\}$, where $\nu$ points out of $D$. The source regards complex function spaces as real Hilbert spaces.

Reading limits: Read the published-paper scan text at printed pp.24–25: equations (1.1) and (1.3), Theorem 1.1, the transposition solution class, and the real inner products at the end of §1. The publisher abstract at https://epubs.siam.org/doi/10.1137/S0363012991223145 corroborates the data/control classes and arbitrary positive time. Scan OCR is imperfect; the proof was not reconstructed and no nonlinear extension is imported.

Geometric control in the presence of a black box · arXiv math/0304184v3, Theorem 1 and (1.1), PDF p.2; §6.2, PDF p.16

For every $T,\varepsilon>0$ and $u_0$ in the printed class $H_0^{1+\varepsilon}(D)$, a control $g\in L^2([0,T]\times\partial D_e)$ drives the linear Dirichlet Schrödinger solution to zero for $t>T$; the obstacle boundary is uncontrolled.

Hypotheses: $\Theta=\bigcup_{j=1}^N\Theta_j$ consists of mutually disjoint closed obstacles with smooth strictly convex boundaries. The smooth bounded outer domain $D_e$ contains $\operatorname{conv}(\Theta)$, and $D=D_e\setminus\Theta$. Section 6.2 requires $\operatorname{conv}(\Theta_j\cup\Theta_k)\cap\Theta_l=\varnothing$ for distinct indices; if $N>2$, also $\kappa L>N$, with $\kappa$ the infimum of principal curvatures and $L$ the minimum obstacle separation (no extra separation inequality for $N=2$).

Reading limits: Read arXiv math/0304184v3 through the unversioned PDF URL, whose first page identifies v3: equation (1.1) and Theorem 1 on PDF p.2, and the obstacle conditions in §6.2 on PDF p.16. Text extraction contains split words. These passages do not define the all-order convention for the printed initial-data class; no spectral/closure identification or proof reconstruction is claimed.

Nicolas Burq, Contrôle de l'équation des plaques en présence d'obstacles strictement convexes · §0.1, printed p.6; Theorem 0.1 and equation (0.1), printed p.7; also Numdam archival scan https://www.numdam.org/article/MSMF_1993_2_55__3_0.pdf, PDF pp.5–6

For every $T,\varepsilon>0$, plate data $(u_0,u_1)\in(H^{1+\varepsilon}(D)\cap H_0^1(D))\times H^{-1+\varepsilon}(D)$ admit $g\in L^2((0,T)\times\partial D_e)$ such that $u_{tt}+\Delta^2u=0$, $u=0$ on all of $\partial D$, and $\Delta u=g$ on the outer boundary (zero on the obstacle boundaries), with $u=0$ after $T$.

Hypotheses: $D_e\subset\mathbb R^3$ is connected and bounded with analytic boundary; mutually disjoint smooth strictly convex closed interior obstacles have convex hull contained in $D_e$, and $D=D_e\setminus\bigcup_j\Theta_j$. Condition H1 is no eclipse by a third obstacle. H2 is $\sum_{\gamma\text{ primitive}}\lambda_\gamma d_\gamma e^{\alpha d_\gamma}<\infty$ for some $\alpha>0$, where $d_\gamma$ is the periodic trapped-ray length and $\lambda_\gamma$ is the square root of the product of the two contracting Poincaré eigenvalues.

Reading limits: Read §0.1 and Theorem 0.1, printed pp.6–7 (PDF pp.5–6), in the publisher sample and Numdam archival text extractions. The OCR of the formulas is imperfect; source-specific plate data notation and the named H1/H2 conditions are retained, without reconstructing their proof or asserting equivalence with the sufficient curvature/separation conditions of Burq–Zworski. No visual inspection credit is claimed.

Sources