Input-output stability degrees for undamped constant coefficients linear partial differential equations
Bibliographic record
Abstract
It is an established fact that systems which have transfer matrices with poles converging to the imaginary axis cannot have exponentially stable time responses. Recently it was proved that for a class of partial differential equations with such a structure of poles it is possible to have a fast decay in time, uniformly in space, as arbitrary polynomials if the initial conditions are smooth enough and with an appropriate decay at infinity. The non-homogeneous version of this result which we present here can be summarized as follows: 'arbitrary regularity in space of the input function leads to arbitrary polynomial convergence in time towards the steady state'. The relation between the properties of the input function and the rate of convergence of the poles to the imaginary axis is quantitative and we indicate methods for computing this rate. We also provide conditions for exponential stability in this context. Due to some limitations in exponential stabilization by feedback a natural alternative to stability enhancement is this polynomial one. Therefore it is useful to investigate when it can be recovered in more practical situations (bounded space, boundary control, etc). Possible applications include the control of distributed oscillatory phenomena (e.g. in large flexible structures, plates), and more recently the control of some advanced materials.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.005 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".