Geometric Jacobian linearization and LQR theory
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Bibliographic record
Abstract
The procedure of linearizing a control-affine system along a non-trivialreference trajectory is studied from a differential geometric perspective. Acoordinate-invariant setting for linearization is presented. With thelinearization in hand, the controllability of the geometric linearization ischaracterized using an alternative version of the usual controllability testfor time-varying linear systems. The various types of stability are definedusing a metric on the fibers along the reference trajectory and Lyapunov'ssecond method is recast for linear vector fields on tangent bundles. Withthe necessary background stated in a geometric framework, linear quadraticregulator theory is understood from the perspective of the Maximum Principle.Finally, the resulting feedback from solving the infinite time optimalcontrol problem is shown to uniformly asymptotically stabilize thelinearization using Lyapunov's second method.
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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 | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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