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CS-278 · computer science · mathematics · Numerical analysis and scientific computing

Stationary and Optimal Linear Differential Behaviors

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

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.

A linear time-invariant differential behavior is a space of trajectories satisfying constant-coefficient linear differential equations. For a polynomial matrix $R$, $\ker R(d/dt)$ denotes trajectories annihilated by its associated differential operator. $\succeq0$ means positive semidefinite.

A state-space system evolves as $\dot x=f(x,u)$ with output $y=h(x,u)$. Model reduction seeks a lower-dimensional system preserving specified input-output properties. Observability distinguishes initial states from their output trajectories, while controllability concerns reachable states under allowed inputs.

Statement recorded in the supplied source.
Characterize the linear time-invariant differential behaviors $\mathcal{B}\in\mathcal{L}^w$ for which there exists a symmetric two-variable polynomial matrix $\Phi\in\mathbb{R}^{w\times w}[\zeta,\eta]$ satisfying $$\mathcal{B}=\ker\Phi\!\left(-\frac{d}{dt},\frac{d}{dt}\right).$$ These are the stationary behaviors. Characterize also the optimal ones, for which one can additionally require $\Phi(-i\omega,i\omega)\succeq0$ for every $\omega\in\mathbb{R}$. [S2]
Scope qualification.
The trajectory regularity and interpretation of the two-variable operator are the report's conventions.
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

Characterize which linear differential behaviors arise as stationary ones and which satisfy the additional frequency-domain positivity condition.

Sources

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 30000200 and source code OWR-792-017 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.

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

CS-278-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on CS-278

Stationary and Optimal Linear Differential Behaviors

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.

A linear time-invariant differential behavior is a space of trajectories satisfying constant-coefficient linear differential equations. For a polynomial matrix $R$, $\ker R(d/dt)$ denotes trajectories annihilated by its associated differential operator. $\succeq0$ means positive semidefinite.

A state-space system evolves as $\dot x=f(x,u)$ with output $y=h(x,u)$. Model reduction seeks a lower-dimensional system preserving specified input-output properties. Observability distinguishes initial states from their output trajectories, while controllability concerns reachable states under allowed inputs.

Statement recorded in the supplied source.
Characterize the linear time-invariant differential behaviors $\mathcal{B}\in\mathcal{L}^w$ for which there exists a symmetric two-variable polynomial matrix $\Phi\in\mathbb{R}^{w\times w}[\zeta,\eta]$ satisfying $$\mathcal{B}=\ker\Phi\!\left(-\frac{d}{dt},\frac{d}{dt}\right).$$ These are the stationary behaviors. Characterize also the optimal ones, for which one can additionally require $\Phi(-i\omega,i\omega)\succeq0$ for every $\omega\in\mathbb{R}$. [S2]
Scope qualification.
The trajectory regularity and interpretation of the two-variable operator are the report's conventions.
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

Characterize which linear differential behaviors arise as stationary ones and which satisfy the additional frequency-domain positivity condition.

Sources

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 30000200 and source code OWR-792-017 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.

Testing a positive normal-equation construction

Given an arbitrary behavior $\ker R(D)$, where $D=d/dt$, a tempting route is to use the formally self-adjoint normal operator $R(-D)^T R(D)$. Its frequency symbol is positive semidefinite. The gap is that equality of kernels on square-integrable vectors does not automatically give equality on the source's entire trajectory space $C^\infty(\mathbb R)$.
Proposition: The scalar behavior $\mathcal B=\{Ce^t:C\in\mathbb R\}$ is not stationary in the source sense. The normal-equation construction applied to $R(z)=z-1$ produces the strictly larger behavior $\operatorname{span}\{e^t,e^{-t}\}$ despite having a nonnegative frequency symbol.
Proof: For a symmetric scalar two-variable polynomial, $\Phi(\zeta,\eta)=\Phi(\eta,\zeta)$. Thus $p(z)=\Phi(-z,z)$ satisfies $p(-z)=\Phi(z,-z)=p(z)$: it is even. If $e^t$ belongs to its differential kernel, then $p(D)e^t=p(1)e^t=0$, so $p(1)=0$. Evenness gives $p(-1)=0$ and hence $p(D)e^{-t}=0$. But $e^{-t}$ is not a constant multiple of $e^t$ on $\mathbb R$. Consequently no such polynomial has kernel exactly $\mathcal B$, including the possibility $p=0$, whose kernel is all smooth trajectories. For the proposed construction choose $\Phi(\zeta,\eta)=(\zeta-1)(\eta-1)$. It is symmetric and gives $p(z)=1-z^2$. The solutions of $w''=w$ are exactly $Ae^t+Be^{-t}$: putting $v=w'-w$ gives $v'=-v$, and solving this first-order equation and then $w'-w=v$ yields the displayed form. At $z=i\omega$, $p(i\omega)=1+\omega^2>0$. This proves all claims and the strict enlargement. ∎
The failed step would have been to integrate a squared residual over the whole real line and infer that a zero normal equation forces zero residual. The source does not require these exponential trajectories to be square integrable, so that inference has no applicable finite-energy hypothesis. Compactly supported variations defining stationarity do not impose square integrability on the trajectory itself.

Verification and remaining gaps

Polynomial multiplication and evaluation at $z=1,-1,i\omega$ were checked. The primary report already discusses the scalar characterization; this example is explanatory auxiliary work, not a new classification. A direct intrinsic characterization for general matrix-valued behaviors and the corresponding optimality conditions was not obtained.

References and source attribution

Primary workshop report: https://doi.org/10.4171/owr/2005/11. Relevant passages read: 2005/11 pp. 639--641, stationary and optimal behaviors and the known scalar case.

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.

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