Inverse Optimal Cardano-Lyapunov Feedback for PDEs with Convection
Bibliographic record
Abstract
We consider the problem of inverse optimal control design for systems that are not affine in the control. In particular, we consider some classes of partial differential equations (PDEs) with quadratic convection and counter-convection, for which the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$L^{2}$</tex> norm is a control Lyapunov function (CLF) whose derivative has either a depressed cubic or a quadratic dependence in the boundary control input. We also consider diffusive PDEs with or without linear convection, for which a weighted <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$L^{2}$</tex> norm is a CLF whose derivative has a quadratic dependence in the control input. For each structure on the derivative of the CLF, we achieve inverse optimality with respect to a meaningful cost functional. For the case where the derivative of the CLF has a depressed cubic dependence in the control, we construct a cost functional for which the unique minimizer is the unique real root of a cubic polynomial: the Cardano-Lyapunov controller. When the derivative of the CLF is quadratic in the control, we construct a cost functional that is minimized by two distinct feedback laws, that correspond to the two distinct real roots of a quadratic equation. We show how to switch from one root to the other to reduce the control effort.
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How this classification was reachedexpand
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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".