Constrained control Lyapunov function construction via approximation of static Hamilton-Jacobi-Bellman equations
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Bibliographic record
Abstract
In this paper, we study the problem of constructing Lyapunov functions for nonlinear input-constrained systems with the largest possible stability regions by using a solution of the associated Hamilton-Jacobi-Bellman (HJB) PDE. To solve this equation, we employ a finite difference approximation and novel boundary conditions based on a recently-developed algorithmic construction of the boundary of the system's null controllable region, which efficiently determines all reachable states. Furthermore, since even smooth HJB PDEs are observed to contain discontinuities, the artificial viscosity perturbation method is used to improve the quality of the approximation. The sub-problem of determining the optimal constrained input at each node is reduced to finding the roots of a certain nonlinear polynomial. Lastly, we illustrate the results using simulation examples.
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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 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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