Lyapunov Functions Certify Stability via Scalar Monotonic Decrease — E8 Intelligence Research

FINDING: Lyapunov functions provide a rigorous, scalar-valued certificate for stability of dynamical systems, reducing high-dimensional convergence to monotonic decrease along trajectories. | MATH: For system \\( \\dot{x} = f(x) \\) with equilibrium \\( x^* \\), a Lyapunov function \\( V: \\mathbb{R}^n \\to \\mathbb{R} \\) satisfies: (i) \\( V(x^*) = 0 \\), \\( V(x) > 0 \\) for \\( x \\neq x^* \\); (ii) \\( \\dot{V}(x) = \\nabla V(x) \\cdot f(x) \\leq 0 \\) (semidefinite) for stability, \\( < 0 \\) for asymptotic stability. Global stability requires \\( V(x) \\to \\infty \\) as \\( \\|x\\| \\to \\infty \\) (radial unboundedness). For discrete maps \\( x_{k+1} = T(x_k) \\), the discrete analogue is \\( V(x_{k+1}) - V(x_k) \\leq 0 \\). In the distributionally robust case (arXiv:2212.01554), one searches for \\( V \\) satisfying \\( \\mathbb{E}_\\xi[V(T(x,\\xi))] - V(x) \\leq -\\gamma \\|x\\|^2 \\) for all distributions within a Wasserstein ball of radius \\( \\varepsilon \\) around the empirical distribution — a min-max SDP. | CONNECTION: T Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com