Characterizing Equilibria in Reach Control Under Affine Feedback**This research is supported by The Natural Sciences and Engineering Research Council of Canada.
Why this work is in the frame
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Bibliographic record
Abstract
The Reach Control Problem (RCP) deals with driving the states of an affine system on a simplex to leave the simplex through a pre-determined facet. A necessary condition for the solvability of the RCP by a given feedback is that there are no closed-loop equilibria in the simplex. As a stepping stone to fully characterizing when equilibria can be removed from the simplex using feedback, this paper studies the geometric structure of open-loop equilibria. Using a triangulation in which the set of potential equilibria intersects the interior of the simplex, we prove that the equilibrium set contains at most one point, in both the single-input and multi-input case. We additionally improve on the currently available results on reach controllability to characterize when the closed-loop equilibria can be pushed off the simplex using affine feedback.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.004 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it