Robust Lyapunov Training Under Model Error

1. Integration point: place the robust penalty in model-based RL or closed-loop sequence training after the learned dynamics model predicts the next state. Use an ensemble of dynamics models or residual statistics to estimate uncertainty. For each replay transition, compute $z=\hat f_{\phi}(x,u)$ and a residual radius $\delta(x,u)$; compute the Lyapunov decrease and add the uncertainty margin before backpropagation into the policy and Lyapunov networks. 2. Pseudocode: fit $M$ dynamics ensembles on the training split; at each batch, evaluate their next-state predictions, set $\delta(x,u)$ to a high quantile such as the empirical 95th-percentile ensemble deviation plus held-out residual error, compute $L_{V}(z)$ either by automatic differentiation $\|\nabla V(z)\|_{2}$ or by a local finite-difference maximum, and minimize the displayed robust loss. Initially stop gradients through $\delta$ to prevent the policy from gaming the uncertainty estimator. Periodically evaluate the robust inequality on held-out transitions. 3. The robust inequality and Lipschitz inflation are the analytic mechanism; $\delta$ and $L_{V}$ are estimated quantities. Calibrate $\delta$ so that at least 95% of held-out residuals fall inside the claimed ball. 4. First cheap experiment: use a two-dimensional nonlinear stabilization system, train with 20%, 50%, and 100% of the transition data, and compare ordinary Lyapunov regularization with uncertainty inflation. Measure the minimum robust margin $m=\min_x[-\Delta V-L_{V}\delta-c_{2}\|x-x^{\star}\|^{2}]$ on held-out states and the fraction of real rollouts that converge. The quantitative prediction is that the robust policy maintains convergence as data are removed until $m$ crosses zero; empirical failures should concentrate near this predicted boundary, while the nominal certificate can remain positive on the learned model but fail on real transitions.