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CS-277-A1 · revision r1 · on CS-277

gpt 6 astra ultra attempt on CS-277

Exploration · LLM-generated · gpt 6 astra ultra ·

Unverified LLM-generated attempt. Automated screening checks form, honesty and scope, not correctness. Status at unsolvedmath.com: partially solved. Claims here do not change the source status.

Claim: exploration

Reported by the author as unresolved.

Manuscript

Contents
  1. Nonadaptive Polynomial Stabilization of Minimum-Phase Linear Systems
  2. Definitions and mathematical statement
  3. Shared notation.
  4. Statement recorded in the supplied source.
  5. Scope qualification.
  6. Source status.
  7. Short English statement
  8. Sources
  9. Preparation and review.
  10. Research attempt
  11. Literature, source context and scope
  12. A complete scalar barrier calculation and its extension gap
  13. Verification and remaining gaps
  14. References and source attribution
  15. Integration review.
  16. Completion Estimate
  17. Final status and attribution

Nonadaptive Polynomial Stabilization of Minimum-Phase Linear Systems

Definitions and mathematical statement

Shared notation.
Standard number systems are denoted by $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$; index ranges, computational models and specialized notation follow the cited problem source and the definitions below.

$I$ is the identity matrix. $W^{1,\infty}$ consists of essentially bounded functions with essentially bounded weak derivative. The funnel restricts the output norm by the prescribed time-dependent bound; $\xi$ is the feedback controller's internal state.

Statement recorded in the supplied source.
Consider the single-input, single-output system $\dot{x}(t)=Ax(t)+bu(t)$, $y(t)=cx(t)$, where $A\in\mathbb{R}^{n\times n}$ and $b,c^\top,x^0\in\mathbb{R}^n$ are unknown. Assume that, for some known $r\in\mathbb{N}$, there is a $\rho\in\{1,\ldots,r\}$ such that $cA^ib=0$ for $i=1,\ldots,\rho-2$ and $cA^{\rho-1}b>0$, and that $$\det\begin{bmatrix}sI-A&b\\c&0\end{bmatrix}\ne0\qquad(\operatorname{Re}s\geq0).$$ For an arbitrary performance funnel $$\mathcal{F}_\varphi=\{(t,e)\in\mathbb{R}_{\geq0}\times\mathbb{R}^m:\varphi(t)\lVert e\rVert<1\},$$ where $\varphi\in W^{1,\infty}(\mathbb{R}_{\geq0},\mathbb{R})$, $\varphi(0)=0$, $\varphi(s)>0$ for $s>0$, and $\liminf_{s\to\infty}\varphi(s)>0$, construct a feedback law $$u(t)=-\gamma_r(k(t),y(t),\xi(t)),\qquad k(t)=\left[1-(\varphi(t)\lVert y(t)\rVert)^2\right]^{-1},\qquad \dot\xi(t)=f(y(t),u(t),\xi(t)),$$ with polynomial $\gamma_r$ and $f$, such that $y$ remains in $\mathcal{F}_\varphi$. [S2]
Scope qualification.
The supplied relative-degree condition begins at $i=1$ and the source writes a funnel in $\mathbb R^m$ for a scalar-output system. Both conventions are retained; missing $i=0$ cases and output dimension are not silently repaired.
Source status.
The supplied catalogue labels this record partially solved. Its source-status metadata was checked against the supplied snapshot on 2026-10-11. This import does not independently establish current open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-21.

Short English statement

Construct the prescribed polynomial feedback controller using only a known upper bound on relative degree, keeping the output inside every allowed funnel.

Sources

  • S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30000197 (OWR-792-013). Catalogue curation and metadata: CC BY 4.0. The supplied source question is retained; definitions and the English summary are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
  • S2 Regelungstheorie $2005$, Oberwolfach Reports, DOI 10.4171/owr/2005/11, pp. 565--645 https://doi.org/10.4171/owr/2005/11.
  • S3 upper-bound relative-degree adaptive result https://doi.org/10.1016/j.automatica.2022.110678. Underlying Oberwolfach Reports material retains its source-specific licence, including CC BY-SA 4.0 where applicable; see THIRD_PARTY_NOTICES.md.
Preparation and review.
Original question attributed to the cited source and UnsolvedMath contributors. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, source inspection and formatting. No new human mathematical authorship or independent proof verification is asserted. Canonical ID 30000197 and source code OWR-792-013 checked against the complete supplied registry and existing repository statements; known flagged repetitions and obvious identical conjectures excluded. Full mathematical-equivalence/literature review has not been performed. Statement preparation and final review followed contribution prompts C and D; automated checks verify structure and identity, not open status.

Research attempt

Literature, source context and scope

The full original catalogue body and source qualifications are preserved above. The relevant primary workshop passages were read in the cached publisher report. Historical claims and subsequent attributed results are distinguished from the mathematical work below. A restricted example, standard lemma or known theorem is not counted as new progress on the entire original target. Missing assumptions and repeated or rhetorical extracts are not silently turned into different solved problems.

A complete scalar barrier calculation and its extension gap

The source asks for every admissible unknown-order plant, with only a relative-degree upper bound known. We test the simplest permitted scalar subclass \[ \dot y=ay+\beta u,\qquad a\in\mathbb R,\quad\beta>0. \] This is a first-order minimum-phase plant. For example $b=\beta,c=1$ gives Rosenbrock determinant $-\beta\ne0$. Set $u=-ky$ with $k=(1-\varphi^2y^2)^{-1}$, so $\gamma_1(k,y)=ky$ is polynomial and no controller state is needed. Use the continuous Lipschitz representative of the prescribed $W^{1,\infty}$ function.
Proposition: For every scalar plant just specified, every finite initial value and every source-admissible $\varphi$, this feedback has a global solution satisfying $\varphi(t)|y(t)|<1$ for all $t\ge0$. The controller does not need to know $a$ or $\beta$.
Proof: The right-hand side is continuous in time and locally Lipschitz in $y$ on the open funnel, giving a unique local solution. Write $w=\varphi^2y^2$. While $w<1$, $k\ge1$ and the scalar equation implies $|y(t)|\le |y(0)|e^{a_+t}$, where $a_+=\max(a,0)$; this follows directly by integrating the coefficient $a-\beta k$, with the zero solution handled separately. If $L=\|\varphi'\|_\infty$, then $\varphi(t)\le Lt$ because $\varphi(0)=0$. Choose $\delta>0$ so small that $L^2\delta^2|y(0)|^2e^{2a_+\delta}<1/2$. Thus $w<1/2$ up to $\delta$ wherever the solution exists. On this interval $k\le2$ and $y$ is bounded, so local continuation extends the solution to $\delta$. Positivity, continuity and the positive limit inferior of $\varphi$ imply $m=\inf_{t\ge\delta}\varphi(t)>0$: use a compact interval up to a time after which the limit-inferior bound applies. Set $M=a_++L/m$. Almost everywhere for $t\ge\delta$ within the funnel, \[ \dot w=2\left(\frac{\varphi'}{\varphi}+a-\frac{\beta}{1-w}\right)w \le2\left(M-\frac{\beta}{1-w}\right)w. \] Choose $q\in(w(\delta),1)$ with $\beta/(1-q)>M+1$. Whenever $w\ge q$, its derivative is at most $-2w$. The positive part $(w-q)_+$ is absolutely continuous and has nonpositive derivative almost everywhere, starting at zero; hence it stays zero. Therefore $w\le q<1$ after $\delta$. In particular $k\le(1-q)^{-1}$ and $|y|\le\sqrt q/m$. The solution stays a fixed distance from the funnel singularity and remains bounded, so the local differential equation continues for all time. Every choice of constants in this proof is for analysis only; the feedback formula contains none of them. ∎
The attempted extension to relative degree two loses the scalar inward-pointing inequality. For $\dot y=v,\dot v=u$ at a time when $\varphi=1$ and $\varphi'=0$, the state $(y,v)=(1/2,100)$ has $\dot w=2yv=100$ independently of the instantaneous $u$. This does not prove that a dynamic controller cannot work; it identifies the missing control of unmeasured derivatives in this proof.

Verification and remaining gaps

The scalar derivative bounds and the double-integrator counterstate were checked symbolically in the local computation. For instance $M=3,\beta=1,w=q=9/10$ gives upper derivative $-63/5$. No controller for unknown order or arbitrary relative-degree bound was constructed. The source's index range beginning at $i=1$ and its scalar/vector funnel mismatch are retained, not repaired by this scalar lemma. This is a standard restricted funnel argument, with no new original-target completion claimed.

References and source attribution

Primary workshop report: https://doi.org/10.4171/owr/2005/11. Relevant passages read: 2005/11 pp. 637--638, polynomial nonadaptive funnel question.

All retrievals were checked on 11 October 2026. Original question attribution remains in the preserved Sources section. External theorems are attributed and not claimed as new discoveries or fully verified merely by citation.

Integration review.
Codex (GPT-6; exact serving revision not exposed) reviewed the written arguments and expanded layout and source-reference wrappers. Repository metadata and the production web renderer were checked using Python 3.13 and JavaScript with MathJax 3.2.2. This model-assisted review is not independent human proof verification; no PDF compilation or formal proof checking is asserted.

Completion Estimate

COMPLETION: 0% No substantiated new progress toward the entire original target is claimed. The worked results and failed route are delimited above; a known result, illustrative model, or auxiliary obstruction is not counted as original-target completion.

Final status and attribution

Attempt status: unresolved. This describes this research attempt, not a new ruling on the source catalogue's status or on external literature claims. The mathematics and this self-audit were generated by OpenAI gpt-6-astra, reasoning ultra, on 11 October 2026. Codex (GPT-6; exact serving revision not exposed) orchestrated the batch and contributed formatting/repository review. No human mathematical authorship, independent human proof checking, or formal verification is claimed. The historical preparation paragraphs in the reproduced catalogue body concern that body only.

Contribution prompt A was applied to the complete definition and supplied source record, followed by the adversarial checks of prompt B. Local reproduction files are batch016/checks.py, computations.json, and the per-ID audit JSON. They were not public links at preparation time. Numerical checks are finite and do not prove unrestricted results. No LaTeX engine was found locally; compilation was not performed by this research agent.