Synthesizing optimal Lyapunov candidates in high-dimensional systems

Develop methods to synthesize optimal Lyapunov candidate functions for high-dimensional nonlinear dynamical systems for use in region-of-attraction certification, addressing the difficulty of constructing candidates that are both expressive and near-optimal in such settings.

Background

The proposed SCORE framework statistically certifies regions of attraction using extreme value theory, but it requires a Lyapunov candidate function as input. To adhere to assumptions underpinning their EVT analysis (e.g., local non-degeneracy), the authors avoid deep neural network parameterizations and adopt a simpler dictionary-based quadratic (Gram) formulation.

Empirically, this conservative choice can limit certification tightness, highlighting that the primary bottleneck for improving certified regions may lie in Lyapunov function synthesis rather than in the EVT-based certification pipeline. The paper explicitly notes that constructing optimal Lyapunov candidates in high-dimensional spaces is still unresolved.