Certified weak-coupling tubes for structured reachability

Research-attempt provenance

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.

For dynamics $\dot x=f(x,u,w)$, controls $u$ and disturbances $w$ determine reachable states. Inner and outer approximations respectively certify membership and contain all possible reachable states. A level-set representation describes a set by a sublevel set of a scalar function. Weak coupling means interactions between components are small in a stated quantitative model; probabilistic verification requires a sampling law and a confidence guarantee.

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. This is a family of numerical-verification questions. The dynamics, coupling scale, horizon, safe set, sampling law, and sense of certification are not fixed; no specific tube theorem is inferred. The supplied catalogue marks this record open; its present research status was not independently checked. The source text below is rendered literally where necessary; damaged formulas and composite extracts are not silently repaired into a new mathematical claim.

Statement recorded in the supplied source.

A.7 Mitchell, Ian

The solution to the Hamilton-Jacobi(-Bellman)(-Isaacs) partial differential equation (HJ PDE) provides the optimal value function (eg cost to go) for bounded input optimal control or zero-sum differential game problems respectively. These equations arise in many disparate fields, and my personal interest lies in how they can be used to approximate the reachable sets of continuous dynamic systems in order to verify and/or synthesize provably safe control policies. For these purposes, it may be possible to relax the goal of a convergent approximation of the solution of the PDE. Instead, we seek guaranteed under or over approximations of the solutions; for example, an underapproximation of the solution might overapproximate the unsafe reachable set, which might thereby generate false negative verifications but no false positives. With or without such relaxations, is it possible to take advantage of structure in the system dynamics to reduce the typically exponential growth in computational cost with system dimension? For example, is it possible to take advantage of weak or linear coupling between certain components to reduce the degrees of freedom needed by an approximation, perhaps through partial parameterization? Or another approach: Can sampling methods be applied to produce quantifiable probabilistic verifications? In addition to these questions related to the application area of verification, I am inter-ested in the class of algorithms designed to approximate solutions of HJ PDEs - often broadly called "level set methods." This interest includes both the time-dependent and stationary (or static) versions of the equation, as well as issues related to the use of such numerical approximations in the control of robotic systems. [S2]

Source status.

The supplied catalogue labels this source record open; 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.

Short English statement

The source asks whether structure or sampling can reduce the cost of certified reachability computations.

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 20001172 (AIM-COMPUTATION-0010). 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 AIM Workshop Problem Lists, source linked by catalogue record AIM-COMPUTATION-0010. https://aimath.org/WWN/optimalcontrol/optimalcontrol.pdf.

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 20001172 and code AIM-COMPUTATION-0010 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.

Literature and context check

The participant contribution in the primary AIM optimal-control workshop PDF was consulted. Its broad scope is retained above. The following explicitly chosen finite or smooth model tests one route suggested by the agenda; it is not a reconstruction of a missing theorem. The elementary arguments are not claimed novel, and no current-resolution audit of the whole field is asserted.

Research attempt: Small coupling can accumulate over unstable horizons

Choose coupled and reference dynamics on the same state space with the same admissible control signals and initial state: $\dot x=f(x,u)+r(x,u)$ and $\dot y=f(y,u)$. Assume both solutions exist on the common time interval being compared. Assume $f$ is uniformly $L$-Lipschitz in state and $\|r\|\le\epsilon$. For the error norm $e$, its upper right derivative satisfies $D^+e\le Le+\epsilon$. Multiplying the scalar comparison equation by $e^{-Lt}$ and integrating gives \[ e(t)\le\epsilon(e^{Lt}-1)/L\quad(L>0),\qquad e(t)\le\epsilon t\quad(L=0). \] One can avoid differentiability of the norm at zero by replacing it with $\sqrt{\|x-y\|^2+\eta^2}$ and letting $\eta\downarrow0$ in the same integral inequality. Matching each admissible signal in both systems gives both directions of the corresponding reachable-set distance bound.

This tube is sharp even in dimension one. Take $f(x)=Lx$, $r=\epsilon$, and both initial states zero. Then $y=0$ and $x=\epsilon(e^{Lt}-1)/L$ exactly. At $T_\epsilon=L^{-1}\log(1/\epsilon)$ for $0<\epsilon<1$, the error tends to $1/L$ as $\epsilon\to0$. Thus "weak coupling" alone does not justify a small-error reduction uniformly over growing horizons or unstable dynamics.

For safety, if a reference reachable set is farther than the displayed radius from the unsafe set, the coupled set is disjoint from it by the triangle inequality. Without that margin, containment in a small tube is inconclusive. Sampling gives a different guarantee: if independent tested inputs follow a fixed law and the unsafe-input probability is $p$, the probability of seeing no failure in $N$ tests is $(1-p)^N\le e^{-Np}$. This bounds a probability under that law, not adversarial safety. An unsafe input in a probability-zero set is never sampled, although it remains an admissible adversarial input. Neither the trajectory tube nor the sampling calculation addresses the cost of representing a high-dimensional set.

Verification and adversarial audit

The exact unstable scalar solution saturates the bound and tests the $L\to0$ limit. Matching controls and identical initial conditions were stated; different admissible signal sets would invalidate the two-sided conclusion. The sampling calculation explicitly assumes independence and a fixed law.

Audit corrections.

Finite-horizon perturbation control was not promoted to uniform long-time accuracy, and sampled confidence was not promoted to worst-case safety.

External computation.

None; symbolic arguments and boundary checks only.

Scope and remaining gaps

The source gives no dynamics, horizon or probability law. The first gap is an approximation and certificate compatible with those actual inputs, including stability and safety margins; no universal dimension reduction or deterministic safety certificate follows from weak coupling or successful samples alone.

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% The calculations below the preserved source statement investigate a concrete route, but establish no substantiated new progress toward the entire original target. Known special cases and failed sufficient conditions are not credited as a solution.

References

The original source references and catalogue attribution are preserved in the Sources section above. The following URLs were actually consulted for this attempt; retention of an additional original reference does not claim fresh consultation of it.

• https://aimath.org/WWN/optimalcontrol/optimalcontrol.pdf (consulted 2026-10-11).

Model and contributor disclosure

OpenAI gpt-6-astra, requested reasoning configuration ultra, generated this research attempt on 2026-10-11. The exact serving revision is not exposed. Codex GPT-6 performed orchestration, repository-format review, and a model-assisted mathematical audit of the written arguments; its exact serving revision is also not exposed. The mathematical route, boundary checks and final adversarial self-audit followed contribution prompts A then B in docs/contribution_prompts.md, after reading CONTRIBUTING.md. Python 3.14.2 assembled the standalone documents; any mathematical computation is specified in Verification. No theorem prover was used. No human mathematical authorship, substantial human proof correction, or independent human proof checking is asserted. Model-assisted review, source retrieval, and structural checks do not establish correctness or novelty. Full private reasoning transcripts are unavailable as reproducibility artifacts; the complete arguments, inputs and outcomes actually used are stated here or in the companion batch audit.

Final status

Unresolved as a new research contribution. This is a concrete route investigation, not a claim of new complete resolution or a current-status certification. A known answer reproduced for context does not become a new contribution. The catalogue's source-status label is preserved; this attempt status does not assert that a source-labelled solved question is open.