Backstepping Control of a Class of Space-Time-Varying Linear Parabolic PDEs via Time Invariant Kernel Functions
Bibliographic record
Abstract
Abstract. This paper aims at addressing the problem of exponential stabilization and continuous dependence of solutions on initial data for a class of 1-D linear parabolic PDEs with space-time-varying coefficients under backstepping boundary control. More precisely, in order to stabilize the system without involving a Gevrey-like condition or the event-triggered scheme, a boundary feedback controller is designed via a time invariant kernel function. Thus, the complexity of control design and implementation can be significantly reduced. The well-posedness of the closed-loop system is assessed by using Schauder’s theory and the [Formula: see text] theory of parabolic PDEs. By using the approximative Lyapunov method and a priori estimates of certain linear operators associated with Volterra integral transformations, the exponential stability of the closed-loop system is established in the spatial [Formula: see text] norm whenever [Formula: see text]. Then, based on the obtained exponential stability, it is shown that the solution to the considered system depends continuously on the spatial [Formula: see text] norm of the initial data. Numerical simulations are conducted to illustrate the effectiveness of the proposed scheme.
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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".