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CS-779 · computer science · mathematics · Artificial intelligence and machine learning · Numerical analysis and scientific computing

Neural Networks for High-Dimensional Optimal Feedback Control

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
Standard number systems are $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$. Specialized notation and index ranges follow the cited source and the definitions below.

Infinite-horizon optimal feedback minimizes the stated accumulated future cost using a state-dependent control. A complexity bound avoids the curse of dimensionality when its dimensional dependence is polynomial rather than exponential at the prescribed accuracy.

Record scope and qualifications.
The catalogue title is retained as metadata. Named certificates, contracts, bounds or frameworks in that title are not asserted as results of the original source extract. The dynamics, running cost, discount or stabilization assumptions, approximation norm, and admissible controls are not specified.

Source-record limitation: The optimal-control dynamics, cost, admissible actions, and accuracy/complexity criteria are absent.

Statement recorded in the supplied source.
Determine when neural networks can solve high-dimensional infinite-horizon optimal feedback-control problems with rigorous accuracy, stability, and complexity guarantees that avoid the curse of dimensionality. [S2]
Source status.
The supplied catalogue labels this source record partially solved; that metadata was checked against the supplied snapshot on 2026-10-11. The label is preserved for provenance and is not an independent verification of present open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-22.

Short English statement

Solve high-dimensional infinite-horizon feedback problems with neural networks and rigorous error, stability, and complexity guarantees.

Sources

  • S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30006098 (OWR-14298803-002). Catalogue curation and metadata: CC BY 4.0. The source extract is retained; definitions, English summaries and qualifications are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
  • S2 Mini-Workshop: Data-driven Modeling, Analysis, and Control of Dynamical Systems (2025), Oberwolfach Reports, DOI 10.4171/owr/2024/57, pp. 3255--3300 https://doi.org/10.4171/owr/2024/57.
  • S3 Nakamura-Zimmerer et al., Adaptive deep learning for HJB https://doi.org/10.1137/19M1288802.
  • S4 Space-time neural HJB method https://doi.org/10.1016/j.cnsns.2025.109196.
  • Source-specific Oberwolfach Reports licences, including CC BY-SA 4.0 where applicable, are retained; see THIRD_PARTY_NOTICES.md.
Preparation and review.
The original source and UnsolvedMath contributors retain attribution for the source extract. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, scope notes and formatting. No new human mathematical authorship or independently verified solution is asserted. Permanent source ID 30006098 and code OWR-14298803-002 checked against the complete supplied registry and current repository. Distinct upstream records, including repeated questions and broad or damaged extracts, are retained for complete coverage of the source CS category. A repeated record is not a new independent conjecture. Complete mathematical-equivalence and current-literature review has not been performed. Preparation and final review follow contribution prompts C and D, with source gaps disclosed. Structural and rendering checks do not establish a new theorem or open status.

Catalogue entry

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

Contributions 1

CS-779-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on CS-779

Neural Networks for High-Dimensional Optimal Feedback Control

Definitions and mathematical statement

Shared notation.
Standard number systems are $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$. Specialized notation and index ranges follow the cited source and the definitions below.

Infinite-horizon optimal feedback minimizes the stated accumulated future cost using a state-dependent control. A complexity bound avoids the curse of dimensionality when its dimensional dependence is polynomial rather than exponential at the prescribed accuracy.

Record scope and qualifications.
The catalogue title is retained as metadata. Named certificates, contracts, bounds or frameworks in that title are not asserted as results of the original source extract. The dynamics, running cost, discount or stabilization assumptions, approximation norm, and admissible controls are not specified.

Source-record limitation: The optimal-control dynamics, cost, admissible actions, and accuracy/complexity criteria are absent.

Statement recorded in the supplied source.
Determine when neural networks can solve high-dimensional infinite-horizon optimal feedback-control problems with rigorous accuracy, stability, and complexity guarantees that avoid the curse of dimensionality. [S2]
Source status.
The supplied catalogue labels this source record partially solved; that metadata was checked against the supplied snapshot on 2026-10-11. The label is preserved for provenance and is not an independent verification of present open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-22.

Short English statement

Solve high-dimensional infinite-horizon feedback problems with neural networks and rigorous error, stability, and complexity guarantees.

Sources

  • S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30006098 (OWR-14298803-002). Catalogue curation and metadata: CC BY 4.0. The source extract is retained; definitions, English summaries and qualifications are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
  • S2 Mini-Workshop: Data-driven Modeling, Analysis, and Control of Dynamical Systems $2025$, Oberwolfach Reports, DOI 10.4171/owr/2024/57, pp. 3255--3300 https://doi.org/10.4171/owr/2024/57.
  • S3 Nakamura-Zimmerer et al., Adaptive deep learning for HJB https://doi.org/10.1137/19M1288802.
  • S4 Space-time neural HJB method https://doi.org/10.1016/j.cnsns.2025.109196.
  • Source-specific Oberwolfach Reports licences, including CC BY-SA 4.0 where applicable, are retained; see THIRD_PARTY_NOTICES.md.
Preparation and review.
The original source and UnsolvedMath contributors retain attribution for the source extract. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, scope notes and formatting. No new human mathematical authorship or independently verified solution is asserted. Permanent source ID 30006098 and code OWR-14298803-002 checked against the complete supplied registry and current repository. Distinct upstream records, including repeated questions and broad or damaged extracts, are retained for complete coverage of the source CS category. A repeated record is not a new independent conjecture. Complete mathematical-equivalence and current-literature review has not been performed. Preparation and final review follow contribution prompts C and D, with source gaps disclosed. Structural and rendering checks do not establish a new theorem or 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.

The source permits discount factor $\beta\in(0,1]$ and explains that favorable separability is needed for useful dimension dependence. We choose its undiscounted case $\beta=1$ and an exactly separable control system. The attempted extension from a small feedback approximation error to a good infinite-horizon controller fails even here unless the error preserves appropriate structure.

Exact dimension-linear feedback and a performance identity

For $d\ge1$, $a\in\mathbb R$, let $\dot x=ax+u$ and \[ J(x_0,u)=\int_0^\infty(\|x(t)\|^2+\|u(t)\|^2)dt, \quad q=\sqrt{a^2+1},\quad p=a+q>0. \] Controls are locally square integrable and trajectories satisfy the ODE; infinite cost is allowed.
Proposition: The optimal feedback is $u=-px$ and the optimal value is $p\|x_0\|^2$. For a scalar gain perturbation $u=-(p+\delta)x$ with $q+\delta>0$, the exact excess cost is \[ J(x_0,u)-p\|x_0\|^2=\frac{\delta^2}{2(q+\delta)}\|x_0\|^2. \] The optimal linear feedback is represented exactly by a ReLU network with $2d$ hidden units.
Proof: The identity $p^2-2ap=1$ gives \[ \|x\|^2+\|u\|^2=\|u+px\|^2-\frac{d}{dt}(p\|x\|^2). \] For any finite-cost control, integrability of $\|x(t)\|^2$ provides a sequence $T_j\to\infty$ with $x(T_j)\to0$. Integrating the identity to $T_j$ and passing to the limit yields $J=p\|x_0\|^2+\int_0^\infty\|u+px\|^2dt\ge p\|x_0\|^2$. The feedback $-px$ has $x(t)=e^{-qt}x_0$ and attains equality. Infinite-cost controls cannot do better.

For the perturbed gain, $x(t)=e^{-(q+\delta)t}x_0$, so direct integration gives $J=(1+(p+\delta)^2)\|x_0\|^2/[2(q+\delta)]$. Subtracting $p\|x_0\|^2$ and using $1+p^2=2pq$ leaves the stated expression. Finally $x_j=\operatorname{ReLU}(x_j)-\operatorname{ReLU}(-x_j)$ realizes each coordinate with two units, followed by multiplication by $-p$. ∎

Uniformly small bias can give infinite cost

In one dimension replace the exact feedback by $\widetilde u(x)=-px+\varepsilon$ with any nonzero constant $\varepsilon$. Its uniform error is exactly $|\varepsilon|$, but the closed-loop trajectory tends to $\varepsilon/q\ne0$. Thus its state-cost integral is infinite, even when $x_0=0$ and the optimal cost is zero. The ODE remains stable toward a shifted equilibrium; the failure is of undiscounted optimal regulation, not of existence of trajectories. Discounting can remove this particular infinite-cost conclusion, so $\beta=1$ is essential.

Verification and remaining gaps

Numerical algebra checks the gain-performance identity for several $a,\delta$ and the ReLU identity. The proof shows that representation size and uniform approximation alone do not establish general optimal feedback guarantees. Nonlinear dynamics, training complexity, certified stability and dimension dependence beyond this separable family remain unresolved.

References and source attribution

Primary workshop report: https://doi.org/10.4171/owr/2024/57. Relevant passages read: OWR 2024/57, printed pages 3278--3280, Grüne on infinite-horizon feedback and separability.

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, read and reviewed the written mathematical arguments, independently reran the finite checks, 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 batch026/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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