From Exponential to Finite/Fixed-Time Stability: Applications to Optimization

The paper proves that scaling an exponentially stable system ẋ = F(x) by σ(x) = η||F(x)||^{-λ} yields global finite-time stability with T(x0) ≤ (2k2L^λ/(k3ηλ))||x0||^λ, and that, under ||F(x)|| ≥ m||x||^β, the scaling σ(x)=η1||F||^{-λ1}+η2||F||^{λ2} yields global fixed-time stability with a uniform bound; both are established via the Lyapunov inequalities k1||x||^2 ≤ V ≤ k2||x||^2 and V̇_F ≤ −k3||x||^2 and standard finite/fixed-time Lyapunov lemmas. The candidate solution reproduces the same key inequalities and integrations. It matches Theorem 1 exactly and derives a valid uniform fixed-time bound by a two-phase threshold argument; the paper uses a direct application of a fixed-time lemma, yielding a slightly different constant (not a substantive discrepancy). Minor omissions in the model (explicit piecewise definition σ=0 at F=0 and uniqueness argument) do not affect correctness.