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